Dubins' Problem on Surfaces. I. Nonnegative Curvature - CMAP
The Journal of Geometric Analysis Volume 15, Number 4, 2005
Dubins’ Problem on Surfaces. I. Nonnegative Curvature By Yacine Chitour and Mario Sigalotti
ABSTRACT. Let M be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any (p1 , v1 ) and (p2 , v2 ) in T M, is it possible to connect p1 to p2 by a curve γ in M with arbitrary small geodesic curvature such that, for i = 1, 2, γ˙ is equal to vi at pi ? In this article, we bring a positive answer to the question if M verifies one of the following three conditions: (a) M is compact, (b) M is asymptotically flat, and (c) M has bounded nonnegative curvature outside a compact subset.
1. Introduction Let (M, m) be a connected, oriented, complete Riemannian manifold and N = U M its unit tangent bundle. Points of N are pairs (p, v), where p ∈ M and v ∈ Tp M, m(v, v) = 1. Given ε > 0, Dubins’ problem consists of finding, for every (p1 , v1 ), (p2 , v2 ) ∈ N , a curve γ : [0, T ] → M, T ≥ 0, parameterized by arc-length such that γ (0) = p1 , γ˙ (0) = v1 , γ (T ) = p2 , γ˙ (T ) = v2 , with geodesic curvature bounded by ε and T as small as possible (depending on (p1 , v1 ), (p2 , v2 )). When the dimension of M is equal to two, Dubins’ problem can be formulated as the time optimal control problem for the following control system, (Dε ) :
q˙ = f (q) + ug(q) , q ∈ N, u ∈ [−ε, ε] ,
where f is the geodesic spray on N (i.e., f is the infinitesimal generator of the geodesic flow on M), g is the smooth vector field generating the fiberwise rotation with angular velocity equal to one and the admissible controls are measurable functions u : J → [−ε, ε], where J is an interval of R. The trajectories of (Dε ) are absolutely continuous curves γ = γu,q (·), with γ the solution of (Dε ) starting at q and associated with the admissible control u. A trajectory γ : [0, T ] → N of (Dε ) is said to be time optimal if, for every trajectory γ % : [0, T % ] → N of (Dε ) such that γ % (0) = γ (0) and γ % (T % ) = γ (T ), we have T ≤ T % . Note that, in the statement of Dubins’ problem, the existence of a curve γ of minimal length is not guaranteed. In the language of control theory, a controllability issue should be solved in
Math Subject Classifications. 14H55, 53C21, 93B05, 93B27. Key Words and Phrases. Controllability, geodesic curvature, nonnegative curvature. Acknowledgements and Notes. This work was partially supported through a European Community Marie Curie Fellowship in the framework of the Control Training Site (CTS), HPMT-CT-2001-00278. 2005 The Journal of Geometric Analysis ISSN 1050-6926
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order to tackle the time optimal problem. Recall that (Dε ) is completely controllable (CC) if, for every q1 , q2 ∈ N , q2 is reachable from q1 , i.e., there exists a trajectory of (Dε ) steering q1 to q2 . For q ∈ N, let Aq ⊂ N be the set of points of N reachable from q.
If M is the Euclidean plane, the dynamics defined by (Dε ) represents, in the robotics literature (cf. [18]), the motion of a unicycle (or a rolling penny) and the projections of the trajectories of (Dε ) on the plane are planar curves parameterized by arc-length with curvature bounded by ε. It is easy to see that (Dε ) is completely controllable for every ε > 0 and, given any pair (p1 , v1 ), (p2 , v2 ) ∈ U R2 , there exists a time optimal trajectory of (Dε ) connecting (p1 , v1 ) and (p2 , v2 ). In 1957, Dubins [8] determined the global structure of time optimal trajectories of (Dε ) in the case where M is the Euclidean plane: He showed that such trajectories are concatenations of at most three pieces made of circles of radius 1ε or straight lines. Further restrictions on the length of the arcs of an optimal concatenation have been proved by Sussmann and Tang [24].
Dubins’-like problems have been proposed by considering more general manifolds M. For instance, the case where M is a two-dimensional manifold of constant Gaussian curvature was investigated in [2, 5, 10, 15] and the case where M = Rn , S n , n ≥ 3 was studied in [16, 17, 23]. Another line of generalization consists of considering the distributional version of (Dε ). For simplicity, suppose M to be two-dimensional. The distributional dynamics can be represented by the two-input control system (DDε ) : q˙ = uf (q) + vg(q) with |u|, |v| ≤ ε (cf. [1]). The controllability issue is trivial since it can be solved infinitesimally: Let h = [f, g], where [., .] denotes the Lie bracket; then the distribution (f, g) is strongly bracket generating, i.e., for every q ∈ N , the triple (f (q), g(q), h(q)) spans Tq N .
In this article, we follow the first path of generalization, i.e., we assume that M is a twodimensional connected Riemannian manifold, oriented and complete (with possibly nonconstant curvature). Our aim is to find geometric or topological conditions on M, such that, for every ε > 0, (Dε ) is completely controllable. We refer to that property as the unrestricted complete controllability (UCC) for Dubins’ problem (we still use the word “Dubins” although we will not consider any optimal control problem). Geometrically, the (UCC) property can be stated as follows: For every (p1 , v1 ), (p2 , v2 ) ∈ N , there exists a curve γ connecting p1 to p2 with prescribed initial and final directions v1 and v2 and with arbitrary small geodesic curvature.
To establish (CC) of (Dε ), ε > 0, we use a standard reduction (cf., for instance, [10]): We will show that (Dε ) is completely controllable if and only if (Dε ) is weakly symmetric, i.e., for every q = (p, v) ∈ N , q − = (p, −v) ∈ Aq . If, for instance, M = R2 , then a control strategy which shows that (Dε ) is weakly symmetric can be given by u so that the resulting trajectory is a teardrop of size 1ε . For R > 0 a teardrop of size R is described in Figure 1. ! then #∗ |N ! → M be a Riemannian covering and (D !ε ) be Dubins’ problem on M; Let # : M ! maps trajectories of (Dε ) onto trajectories of (Dε ). Therefore, if the (CC) property holds for !ε ), then it also holds for (Dε ). Equivalently, if (D !ε ) is weakly symmetric, then (Dε ) is. For (D instance, if M is flat, then a controllability strategy for M is obtained by projecting the one of the Euclidean plane, seen as the universal covering of M. This simple idea of applying strategies which are valid on a Riemannian covering of the manifold (not necessarily a universal covering) will be repeatedly exploited in the article. The first condition ensuring (UCC) which we obtain is purely topological: If M is compact, then, by means of a Poisson stability argument, (UCC) turns out to hold for Dubins’ problem. Thus, we are let to the case where M is noncompact. The geometric quantity which plays a crucial role in the characterization of controllable Dubins’ problems is the Gaussian curvature of M, denoted by K. The curvature appears quite soon in the study of the Lie bracket configuration defined by (Dε ) and, therefore, of local controllability
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
FIGURE 1
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The teardrop trajectory of size R.
issues: Indeed, for every q ∈ N , [f, [f, g]](q) = −K(π(q))g(q) ,
(1.1)
where π : N → M denotes the bundle projection. The role of K is reinforcedly suggested by the following fact: If M is the Poincaré half-plane (K ≡ −1), then (Dε ) is completely controllable if and only if ε > 1 (cf. [5, 15]). Roughly speaking, this happens because the negativeness of K not only prevents the geodesics to have conjugate points but actually is an obstacle for the controlled turning action to overcome the spreading of the geodesics. When |u| ≤ ε ≤ 1, the steering effect of u g is not strong enough and (CC) fails to hold. It is therefore natural to formulate necessary conditions for (UCC) in terms of the Gaussian curvature K. For instance, an extension of the case K ≡ 0 is given by the situation in which M is asymptotically flat, i.e., K tends to zero at infinity. Under this hypothesis, we are able to prove the (UCC) property: The control strategy is based on the possibility of tracking a teardrop loop in a covering domain over a piece of M at infinity. We will see that a suitable covering manifold can be globally described by a single appropriate geodesic chart. Bearing in mind the previous example of noncontrollability, it is reasonable to study first the situation where the curvature is nonnegative. The negative curvature case is the subject of subsequent article [22], where we provide a complete characterization of two-dimensional nonpositively curved manifolds M, with either uniformly negative or bounded curvature, that satisfy property (UCC). Such characterization involves the limit set of M as well as an integral decay condition of K at infinity. In the nonnegative curvature case, the one treated here, no local spreading effect due to the drift term has to be compensated. A result by Cohn-Vossen (cf. [6]) implies that, if K ≥ 0 and K is not identically equal to 0, then M is diffeomorphic to a plane and, more importantly for the
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controllability issue, "
M
K dA ≤ 2π ,
(1.2)
where dA is the surface element in M. (It is actually true that, in any dimension, a complete noncompact Riemannian manifold whose sectional curvatures are all everywhere nonnegative and with one point where they all are positive is diffeomorphic to an Euclidean space, as proved by Perelman in [19].) As a consequence of (1.2), for any fixed radius R > 0, the total curvature on the disk centered at p with radius R tends to zero as p tends to infinity. The same is true under the relaxed hypothesis that K is nonnegative outside a compact subset of M. The integral decay of K to zero can be interpreted as a kind of asymptotic flatness condition and it suggests that (Dε ) should be completely controllable for every ε > 0. We are able to confirm that intuition under the additional assumption that K is bounded over M, i.e., K∞ = supM K is finite. The existence of a control strategy which allows to track a teardrop loop at infinity is much more delicate to prove than in the asymptotically flat case. The key step is the identification of a simply connected covering domain on which the teardrop strategy can be applied. The covering domain D is not described anymore by one single chart, as in the asymptotic flat case, but by gluing rectangular strips, each of them obtained by one regular geodesic chart. There are O( 1ε ) such strips and each of them has width proportional to √K1 . Then, using these strips, one is able to mesh D by ∞
geodesic quadrilaterals Pj,k with edges of length proportional to
√1 . K∞
The tracking operation is
now decomposed in O( 1ε ) steps: We design a discrete approximation of the teardrop, by fixing a sequence of O( 1ε ) points on the edges of the polygons Pj,k and by associating with each of them a corresponding direction. After that, we solve the problem of connecting pairs of subsequent points of the approximating sequence by an admissible trajectory, being tangent to the associated directions. Each elementary problem of this kind can be formulated in a single coordinate strip. Intuitively, its solution is based on the topological description of small time attainable sets for nondegenerate single-input control-affine three-dimensional systems, due to Lobry [13]: The set of points which are reachable from q0 ∈ N in small time, is given by the region enclosed by two surfaces, obtained as union of all small-time bang-bang trajectories from q0 with one switch. What we do in practice, is to estimate the coordinate expression of such surfaces and to check whether they enclose the final state of the elementary problem. The article is organized as follows. In Section 2, we gather the notations used in the article, describe the general construction of local covering domains, establish basic properties for Dubins’ problem and, finally, study the case where M is compact. Section 3 is devoted to the asymptotically flat case, where the Gaussian curvature tends to zero at infinity. In Section 4, the unrestricted complete controllability property is established when K is nonnegative and bounded outside a compact domain.
2. Basic notations and first results 2.1. Differential geometric notions Let (M, m) be a complete, connected, oriented, two-dimensional Riemannian manifold. Denote by K its Gaussian curvature and by N the unit tangent bundle U M. Let π : N → M be the canonical bundle projection of N onto M. We will usually denote by p a point in M and by q = (p, v) one in N, where p = π(q) and v ∈ Tp M, m(v, v) = 1. Given v ∈ Tp M, we write v ⊥ for its counterclockwise rotation in Tp M of angle π/2. For every q = (p, v) ∈ N , we set q ⊥ = (p, v ⊥ ) and q − = (p, −v).
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Given p1 , p2 ∈ M, d(p1 , p2 ) denotes the geodesic distance between p1 and p2 . When no confusion is possible, we simply write -p- (respectively, -q-) to denote the distance d(p, p0 ) (respectively, d(π(q), p0 )) from a fixed point p0 ∈ M.
Let f be the geodesic spray on T M, whose restriction to N (still denoted by f ) is a welldefined vector field on N. Recall that f is characterized by the following property: p(·) is a geodesic on M if and only if (p(·), p(·)) ˙ is an integral curve of f . In particular, f satisfies the relation π% (f (q)) = q .
(2.1)
Denote by g the smooth vector field on N, whose corresponding flow at time t is the fiberwise rotation of angle t. In terms of the covariant derivative on M, the integral curves of f and g are solutions, respectively, of the following equations, # # p˙ = 0 , p˙ = v , and ∇v v = 0 , ∇v v = v ⊥ . We write etf (respectively, etg ) to denote the flow of f (respectively, g) at time t. If x0 belongs to a metric space (X, d0 ) and ρ > 0, then Bρ (x0 ) denotes the open ball of center x0 and radius ρ. Given a subset Y of X, Clos(Y ) and Int(Y ) are, respectively, the closure and the interior of Y . In the sequel of the article, we will systematically use as local coordinates the geodesic ones, whose definition is recalled below. Its construction has a crucial role in the present exposition, since it allows to define a wide class of local covering domains of M. Given q ∈ N, consider the map R2 (x, y)
φq :
−→ /−→
M $ % π π eyf e 2 g exf (q) .
Fix R = [x1 , x2 ] × [y1 , y2 ] ⊂ R2 and assume that the origin (0, 0) belongs to R. If φq is a local diffeomorphism at every point of R, then R can be endowed with the Riemannian structure lifted from M, in such a way that φq becomes a local isometry. If this happens, we denote by R(q) the manifold with boundary which is obtained. The segment [x1 , x2 ] × {0}, which is the support of a geodesic in R(q), is called the base curve of R(q). The Gaussian curvature of R(q) at a point (x, y) is given by K(φq (x, y)), and, where no confusion can arise, will be denoted by & & K(x, y). If R is a neighborhood of (0, 0) and φq &R is injective, then φq &R is a geodesic chart on M. In the coordinates (x, y), m has the form m(x, y) = B 2 (x, y) dx 2 + dy 2 ,
where B : R → R is the solution of the system B(x, 0) ≡ 1,
By (x, 0) ≡ 0,
and
Byy + KB = 0 ,
(2.2)
in which the index y appearing in By , Byy stands for the partial differentiation with respect to y. (See, for instance, [11].) Notice that, for every point q ∈ N and every small enough rectangular neighborhood R & of (0, 0), φq &R is a geodesic chart on M. In general, if B is the solution of (2.2) on R, with
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K = K ◦ φq , then R(q) is well defined if and only if B is everywhere positive on R. In R(q), define the real-valued function F by F (x, y) =
By (x, y) . B(x, y)
(2.3)
The unit bundle U R(q) can be identified with & ( ' & (x, y, vx , vy ) ∈ R4 & (x, y) ∈ R, B 2 (x, y)vx2 + vy2 = 1 .
Equivalently, a set of coordinates in U R(q) is given by (x, y, θ ) ∈ R × S 1 , with the identification Bvx = cos θ,
vy = sin θ .
(2.4)
Remark 2.1. Notice that, for all points (x, y, θ ) in U R(q) such that y = 0, the coordinate θ
measures the Riemannian angle between the corresponding unit vector and the base curve of R(q). On the other hand, a unit vector of coordinates (x, y, 0) or (x, y, π ) is always φq∗ m-orthogonal to the segment {x} × [y1 , y2 ].
More generally, the function which associates with θ the angle between (x, y, 0) and (x, y, θ ) is a Lipschitz continuous map from S 1 into itself, with Lipschitz constant continuously depending on B(x, y). In geodesic coordinates, f and g are given by ) *T cos θ f (x, y, θ ) = , , sin θ, F (x, y) cos θ B(x, y)
g(x, y, θ ) = (0, 0, 1)T .
(2.5)
The pair of vector fields (f, g) define a contact distribution on N , i.e., the triple (f (q), g(q), [f, g](q)) spans Tq N for every q ∈ N, where [·, ·] stands for the Lie bracket. The Lie-algebraic structure of the contact distribution {f, g} is characterized by the relations (i) [f, g] = h ,
(ii) [g, h] = f ,
(iii) [h, f ] = Kg ,
where h, defined by (i), is represented in geodesic coordinates as *T ) sin θ , − cos θ, F (x, y) sin θ . h(x, y, θ ) = B(x, y)
(2.6)
(2.7)
A proof of (2.6) can be obtained, for instance, by using the expressions (2.5) of f and g in geodesic coordinates. Equivalently, (2.6) could have been derived from the structure equations arising from the moving frame approach (see [5]). A metric m ! on N can be introduced by requiring that (f (q), g(q), h(q)) is an m !-orthonormal basis of Tq N , for every q ∈ N . Such m ! is usually called the Sasaki metric inherited from m and endows N with a complete Riemannian structure. (See, for instance, [20].)
2.2. The control system Recall that, for every ε > 0, (Dε ) denotes the control system (Dε ) :
q˙ = f (q) + ug(q) ,
q ∈ N,
u ∈ [−ε, ε] .
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By definition, an admissible control is a measurable function u(·), defined on some interval of R, with values in [−ε, ε]. The solutions of (Dε ) corresponding to admissible controls are called admissible trajectories. It follows from (2.1) and the definition of g that, for every admissible trajectory q : [0, T ] → N of (Dε ), d(π(q(0)), π(q(T ))) ≤ T . Therefore, M being complete, for every admissible control u : R → [−ε, ε], the non-autonomous vector field f + u(t)g is complete, that is, with any initial condition q0 ∈ N we can associate a solution q(·) of (Dε ), defined on the whole real line, such that q(0) = q0 . In other words, the control system (Dε ) is complete. For every q ∈ N and T > 0, the attainable set from q within time T is the set ATq consisting of the endpoints of all admissible trajectories starting from q and defined on a time interval of length smaller than T . Similarly, the attainable set from q is the set Aq consisting of the endpoints of all admissible trajectories starting from q. The control system (Dε ) is called completely controllable if Aq = N for every q ∈ N.
Definition 2.2. We say that Dubins’ problem on M has the unrestricted complete controllability (UCC) property if, for every ε > 0, (Dε ) is completely controllable. In local geodesic coordinates, (Dε ) can be written as follows, x˙
y˙ θ˙
=
= =
cos θ , B sin θ , u + F cos θ .
More intrinsically, we can rewrite system (2.8)–(2.10) in the form # p˙ = v , ∇v v = uv ⊥ ,
(2.8) (2.9) (2.10)
(2.11)
which accounts for a clear geometric interpretation of the unrestricted controllability property: The Dubins problem on M is unrestrictedly completely controllable if and only if, for every (p1 , v1 ), (p2 , v2 ) ∈ N , for every ε > 0, there exists a curve p : [T1 , T2 ] → M with geodesic curvature smaller than ε such that p(Ti ) = pi , p(T ˙ i ) = vi , i = 1, 2.
The fact that f and g define a contact distribution on N has the important consequence that, for every 0 < t < T and q ∈ N, etf (q) belongs to Int(ATq ). This follows, for instance, from the description of small-time attainable sets for single-input nondegenerate three-dimensional control systems given by Lobry in [13]. From the viewpoint of control theory, the property that the distribution defining (Dε ) has a contact structure implies that (Dε ) is bracket generating, i.e., such that the iterated Lie brackets of f and g span the tangent space to N at every point.
Remark 2.3. If q : [0, T ] → N is a trajectory of (Dε ) corresponding to some admissible
control u : [0, T ] → [−ε, ε], then the trajectory q(T − ·)− obtained from q(·) by reflection and time-reversion is itself an admissible trajectory of (Dε ) and steers q(T )− to q(0)− . Its corresponding control function is given by −u(T − ·), which is indeed admissible. Therefore, for every q, q % ∈ N , q % belongs to Aq if and only if q − belongs to A(q % )− .
Remark 2.4. Assume that, for every q in N , q − ∈ Aq , i.e., that (Dε ) is weakly symmetric.
Then, due to Remark 2.3, q % ∈ Aq if and only if Aq = Aq % . It follows that, for every q ∈ N and
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every q % ∈ Aq ,
% $ % $ % $ q % ∈ Int Ae−tf (q % ) = Int Aq % = Int Aq ,
where t > 0 and the first inclusion follows from the previously quoted Lobry’s result. Therefore, {Aq }q∈N is an open partition of N . Since N is connected, then (Dε ) is completely controllable.
Thanks again to Remark 2.3, we obtain the following equivalence: (Dε ) is completely controllable if and only if, for every q ∈ N , there exists q % ∈ Aq such that (q % )− ∈ Aq % .
Remark 2.5. If M is nonorientable, then the vector field g is not well defined. Anyhow, the control problem still makes sense, since locally g can be defined fixing arbitrarily an orientation, and the system is independent of this choice. Formally, Dubins’ problem can be defined as a control-affine system with multiple controls, using a partition of unity on M in order to glue the local definitions of g together. It is known that M admits an oriented Riemannian double ! → M. Since the additional hypothesis under which we get the (UCC) property covering # : M (cf. Proposition 2.7 and Theorems 3.1, 4.1) are shared by any finite Riemannian covering, then the results of this article extend to nonorientable manifolds. A sufficient condition for unrestricted complete controllability is the compactness of M. This fact is a consequence of a more general result on controllability on compact manifolds of bracket generating systems made of conservative vector fields due to Lobry [14]. Lemma 2.6 gives a stronger formulation of Lobry’s result, adapted to the specific control system (Dε ), which implies also that every attainable set is unbounded when M is open. The proof is a variation on the classical one of Poincaré’s theorem on volume-preserving flows.
Lemma 2.6. If q ∈ N exists such that Aq is relatively compact in N, then M is compact and Aq = N.
Proof of Lemma 2.6.
Fix q ∈ N and assume that G = Clos(Aq ) is compact in N. As already remarked, for every t > 0 and every q % ∈ N, etf (q % ) ∈ Int(A2t q % ). Fix t = 1. The compactness of 2 % G and the continuous dependence of Aq % on q imply that there exists ρ > 0 such that, for every q % ∈ G, $ $ %% Bρ ef q % ⊂ Aq % .
(2.12)
We want to prove that ∂Aq is empty. Let, by contradiction, r ∈ ∂Aq . A well-known theorem by Krener [12] states that any attainable set of a bracket generating system is contained in the closure of its interior. Therefore, V = Aq ∩ Bρ (r) has nonempty interior and, in particular, its volume is strictly positive. Since ef is a volume preserving diffeomorphism of N (see, for instance, [20]) and Aq has finite volume (it is bounded), then {enf (V )}n∈N cannot be a disjoint family, being enf (V ) ⊂ Aq for every n ∈ N. Therefore, there exist n1 < n2 such that en1 f (V ) ∩ en2 f (V ) is not empty. Equivalently, there exists a point in e(n2 −n1 −1)f (V ) whose image by ef lies in V . Due to (2.12), it follows that r ∈ Int(Aq ) and the contradiction is reached.
Proposition 2.7. Let M be a complete, connected, oriented, two-dimensional Riemannian manifold. Assume, in addition, that M is compact. Then Dubins’ problem is unrestrictedly completely controllable. For the rest of the article, we deal with the case M noncompact.
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3. Asymptotically flat manifolds Throughout this section, we assume that M is asymptotically flat, that is, lim K(p) = 0 .
-p-→∞
(3.1)
For every L > 0, let QL = [0, 2L] × [−L, L]. According to the notation introduced in Section 2.1, if the map φq0 , q0 ∈ N , is a local diffeomorphism at every point of QL , then QL (q0 ) denotes the Riemannian manifold (with boundary) obtained endowing QL with the Riemannian structure lifted from M. Let us characterize values of L for which the construction of QL (q0 ) can be carried out. Let B be the solution of (2.2) on QL , with K = K ◦ φq0 . Set & & (3.2) δ = max &K ◦ φq0 & . QL
By Sturm-Liouville theory, we can compare B with the solution of (2.2) √ √ corresponding to K constantly equal to δ. We obtain that, if δ|y| ≤ π2 , then B(x, y) ≥ cos( δy) ≥ 0 for every x ∈ [0, 2L]. Thus, if π L< √ , 2 δ
(3.3)
then QL (q0 ) is well defined. In particular, since M is asymptotically flat, then, for every L > 0 and every q0 outside a compact subset of N (depending, in general, on L), QL (q0 ) is well defined. We stress that no global finiteness property is stated (nor needed) for the projection φq0 from QL (q0 ) onto its image. In general, for L fixed, the cardinality of the set of preimages φq−1 (q0 ) 0 can fail to have a uniform bound when q0 varies in N . The situation will be different in Section 4. Together with m, also the control problem (Dε ) is lifted from N to U QL (q0 ). Let us stress the trivial, but crucial, property that every admissible trajectory of the lifted control system is projected by φq0 to an admissible trajectory of (Dε ). In the coordinates (x, y, θ ) of U QL (q0 ), the dynamics of the lifted system is described by (2.8)–(2.10). Due to Remark 2.4 and Lemma 2.6, the proof of the complete controllability of (Dε ) reduces to show that q0− ∈ Aq0 if δ is small enough. This will be done by designing an admissible trajectory for the lifted control problem on U QL (q0 ), steering (0, 0, 0) to (0, 0, π). Fix q0 ∈ N , L > 0 and assume that
√ π δ≤ , 3L
(3.4)
where δ is defined as in (3.2). The Sturm-Liouville theory, together with the well definedness of QL (q0 ), implies that $√ % $√ % cos δy ≤ B(x, y) ≤ cosh δy , (3.5) and
$√ % & & & By (x, y) & √ sinh δ|y| & & |F (x, y)| = & ≤ δ $√ % , B(x, y) & cos δy
(3.6)
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for every (x, y) ∈ QL (q0 ). An upper bound for |F | in QL (q0 ) is given by we can assume that ε max |F | ≤ , QL (q0 ) 2
√
√
δ sinh(√ δL) . Then, cos( δL)
(3.7)
by taking √ δ≤
ε $ %. 4 sinh π3
(3.8)
Consider now the control system (Dε/2 ) on the unit bundle of the Euclidean plane. Let u(·) be the control function corresponding to the trajectory whose projection on R2 is a teardrop of size 2/ε which leaves the origin horizontally and arrives at the origin with the opposite direction (Figure 1, with R = 2/ε). Thus, u(·), is piecewise constant, taking alternately the values −ε/2 and ε/2. Denote the coordinates of the teardrop trajectory in R2 × S 1 by x(·), y(·) and θ(·). It √follows from straightforward computations that (x(·), y(·)) takes values in the rectangle [0, 2( 3 + 1)/ε] × [−2/ε, 2/ε] and that the teardrop has length 14π 3ε . Fix L=
3 . ε
The idea is to apply to the lifted system the time-variant feedback control u(t) = u(t) − F (x, y) cos θ ,
(3.9)
which is admissible, as long as the corresponding trajectory stays in U QL (q0 ), since (3.7) holds. Consider the solution γ (·) = (x(·), y(·), θ (·)) of (2.8)–(2.10) corresponding to u(·), with initial condition γ (0) = (x0 , 0, 0). As long as (x(t), y(t)) stays in QL , we have y(t) = y(t) and θ (t) = θ (t). Therefore, & & " t & & 1 |x(t) − x(t) − x0 | ≤ | cos(θ (s))| && − 1&& ds B(x(t), y(t)) 0 & & & & 14π 1 ≤ max && − 1&& . 3ε QL (q0 ) B $ % It follows from (3.5) that, for every α ∈ 0, π2 , if √ δL ≤ α , (3.10) then & & &1 & cosh(α) − 1 & max & − 1&& ≤ . QL (q0 ) B cos(α)
Therefore, it is possible to fix α, independent of ε, such that, whenever δ satisfies (3.10), & & &1 & 14π 1 max && − 1&& ≤ . 3ε QL (q0 ) B 4ε
1 Assume that (3.10) is satisfied and fix x0 = 4ε . Then γ (·) is defined for the entire time duration of u(·). At its final point, its coordinates are of the type (x1 , 0, π). Concatenating γ with two trajectories corresponding to control equal to zero, we obtain an admissible trajectory
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
575
for Dubins’ problem lifted to U QL (q0 ), steering (0, 0, 0) to (0, 0, π). We proved the following theorem.
Theorem 3.1. Let M be a complete, connected, oriented, two-dimensional Riemannian manifold. Assume, in addition, that M is asymptotically flat. Then, Dubins’ problem is unrestrictedly completely controllable. Actually, from the nature of the above argument, a stronger result follows.
Proposition 3.2. There exists a universal constant µ > 0 such that, if lim sup-p-→∞ |K(p)| ≤ µε2 , then (Dε ) is completely controllable.
Proposition 3.2 can be recovered from the smallness conditions (3.4), (3.8), and (3.10) imposed on δ, where L should be replaced by 3/ε and α can be given explicitly.
4. Manifolds with nonnegative curvature outside a compact set 4.1. Construction of the covering domain From now on, in addition to the general assumptions on M + made in Section 2, we will assume K to be nonnegative outside a compact subset of M. Since M K dA, the total curvature of M, is well defined (allowing extended values), and larger than −∞, then it follows from a result by Huber [9], that M is finitely connected. Therefore, Cohn-Vossen theorem [6] applies, i.e., " K dA ≤ 2π χ (M) , (4.1) M
where χ (M) is the Euler characteristic of M. The main result of the section is the following.
Theorem 4.1. Let M be a complete, connected, oriented, two-dimensional Riemannian manifold. Assume , in addition, that the Gaussian curvature K of M is bounded and {p ∈ M| K(p) < 0} is relatively compact in M. Then, Dubins’ problem is unrestrictedly completely controllable. In the sequel, we assume that K is bounded on M and we set K∞ = sup K(p) . p∈M
Without loss of generality, K∞ > 0. than
For every L, d > 0, let QL,d = [−L, L]×[−d, d]. As remarked in Section 3, if d is smaller √π , then the local covering domain QL,d (q0 ) is well defined. Fix 2 K∞
π . d= √ 3 K∞
(4.2)
"
(4.3)
We claim that, for every L > 0, lim
-q-→∞ QL,d (q)
K dA = 0 .
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Yacine Chitour and Mario Sigalotti
The result follows from general properties of Riemannian surfaces. Recall that the injectivity radius at a point p of M is defined as the least upper bound of all R > 0 such that the exponential map expp : Tp M −→ M , restricted to the disk BR (0), is injective. It is denoted by ip (M), while i(M) = inf p∈M ip (M) is called the injectivity radius of M. Property (4.3) is proved as soon as we show that i(M) > 0, since, due to Cohn-Vossen theorem, for every R > 0, lim
"
-p-→∞ BR (p)
K dA = 0 .
(4.4)
Lemma 4.2. Let M be a complete, connected, two-dimensional Riemannian manifold. Assume that K is bounded on M and nonnegative outside a compact subset. Then i(M) > 0. We were not able to find in the literature an explicit statement of Lemma 4.2, although it happens to be a rather straightforward consequence of known facts. It worth mentioning, anyhow, that, in the special case when 0 ≤ K ≤ K∞ , a positive lower bound on the injectivity radius of M is known. Indeed, a theorem √ by Sharafutdinov [21, 7] states that i(M) is larger than or equal to the minimum between π/ K ∞ and the injectivity radius of the Cheeger-Gromoll soul of M, which is a compact submanifold of M. In the nonflat two-dimensional case, the soul is always atomic and its injectivity radius can be set to be +∞.
Proof of Lemma 4.2.
A general result [4, Lemma 5.6], which holds for any complete Riemannian manifold, states that, for every p ∈ M, ip (M) = min{t > 0 | t is a conjugate time for a geodesic γ : [0, +∞) → M, γ (0) = p , or 2t is the length of a geodesic loop passing through p} . (4.5)
Fix a compact set M0 of M such that {p ∈ M| K(p) < 0} ⊂ M0 . We can assume that ClosM \ M0 is a finite union of tubes of M, that is, according to the definition of Busemann [3], subsets of M which are homeomorphic to half-cylinders and whose boundaries are simple closed geodesic polygons. Moreover, due to Cohn-Vossen theorem, we can suppose that the total curvature of each tube is smaller than π. Fix one of such tubes and denote it by T . We have to prove that inf p∈T ip (M) > 0. Let γ be a simple geodesic loop contained in the interior of T . Then γ identifies two connected regions of T , a bounded and an unbounded one. The bounded region B must contain the boundary of T , otherwise Gauss-Bonnet theorem would constrain the total curvature of B to be greater than or equal to π, contradicting the assumptions made on T . Property (44.16) in [3] (originally stated in the general framework of G-surfaces) implies that there exists a curve in T , freely homotopic to ∂T , of minimal length. In particular, the length of γ is bounded from below by a positive constant. On the other hand, since K ≤ √ K∞ , the first conjugate time for any half geodesic contained in M is bounded from below by π/ K∞ . The lemma follows from (4.5). Let us show how the integral smallness of K on the covering domains QL,d (q), ensured — at infinity — by (4.3), allows to estimate the evolution of geodesics in coordinates. Let p(·) be a geodesic in QL,d (q), parameterized by arc-length; then (p(·), p(·)) ˙ is a curve
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
577
in U QL,d (q), whose coordinates satisfy x˙
=
y˙ θ˙
= =
cos θ , B sin θ , F cos θ ,
(4.6) (4.7) (4.8)
which is a particular case of (2.8)–(2.10). Denote by σz the curve in U QL,d (q), which is solution of the system (4.6)–(4.8) with initial condition σz (0) = z ∈ U QL,d (q), and let [−T1,z , T2,z ] be its maximal interval of definition. Let
, $ % N 0 = q ∈ N | d π(q), K −1 ((−∞, 0)) > L + d ,
and remark that the complement of N 0 in N is compact. The set N 0 is defined in such a way that, for every q0 ∈ N 0 , K is nonnegative on QL,d (q0 ). Fix
π 4
0 verify Q K dA ≤ δ ≤ δ0 , then, for every z0 = (0, y0 , θ0 ) with . / y0 ∈ [− d2 , d2 ] and θ0 ∈ − θ , θ , the corresponding σz0 (·) = (x(·), y(·), θ (·)) satisfies (i)
for every t in [−T1,z0 , T2,z0 ],
|x(t) − cos(θ0 )t|
|y(t) − y0 − sin(θ0 )t| |θ (t) − θ0 |
Moreover, if |θ0 | ≤ (ii)
d 8L ,
≤
≤ ≤
(2L + d)δ , 2Lδ , δ.
(4.9) (4.10) (4.11)
then
x(−T1,z0 ) = −L and x(T2,z0 ) = L.
Proof of Lemma 4.3. every (x, y) ∈ Q,
As it was done for (3.5) in Section 3, one obtains from (4.2) that, for 1 ≤ B(x, y) ≤ 1 . 2
(4.12)
Fix z0 = (0, y0 , θ0 ) and σz0 (·) = (x(·), y(·), θ (·)) as in the statement of the lemma. Let [−T1 , T2 ] = [−T1,z0 , T2,z0 ] and denote by (−t1 , t2 ) the maximal open neighborhood of zero (in [−T1 , T2 ]) such that, for every t ∈ (−t1 , t2 ), |θ (t)|
0, for every q outside a compact subset of N (depending, in general, on δ), " and K dA ≤ δ . (4.16) K ≥ 0 on QL,d (q) QL,d (q)
We can choose δ ≤ δ0 (defined as in Lemma 4.3), and fix q0 ∈ N such that (4.16) holds for every q verifying d(π(q), π(q0 )) ≤ 2L. We next build a Riemannian two-dimensional manifold, which will be a finite covering of a region of M containing π(q0 ). The covering domain will be obtained by gluing together several rectangles of the type QL,d (q). The purpose is to track a teardrop of size r/ε, r > 1 to be fixed, for the lifted Dubins problem and to eventually obtain that q0− ∈ Aq0 . In this perspective, since L will measure the size of the covering domain, we fix L=
4r . ε
(4.17)
The (UCC) property will follow, as in the asymptotic flat case, from the fact that the tracking operation can be performed for any q0 outside a big enough compact subset of N (depending, in general, on ε). From now on, we assume, sometimes implicitly, that δ is as small as needed with respect to ε, 1/r and 1/K∞ . Notice that r is considered independent of ε and K∞ . We will denote by C(·) any constant which is a function of the quantities appearing in its argument. For instance, C(K∞ , ε, r) denotes a constant which depends on K∞ , ε, and r. Fix y=
d . 8
(4.18)
For every q verifying (4.16), let σ (q, ·) = (x(q, ·), y(q, ·), θ (q, ·)) be the solution in U QL,d (q) of the system (4.6)–(4.8), with initial condition (x(q, 0), y(q, 0), θ (q, 0)) = (0, −y, 0) . Denote by [−T1 (q), T2 (q)] the maximal interval of definition of σ (q, ·). If δ is small enough, then we can assume that y(q, ·) takes only negative values. Let W (q) be the region of QL,d (q) defined by W (q) = {(x(q, s), t)| s ∈ [−T1 (q), T2 (q)], t ∈ [y(q, s), 0]} ,
(4.19)
whose boundary is given by [−L, L] × {0}, {−L} × [y(q, −T1 (q)), 0], {L} × [y(q, T2 (q)), 0], and by 1(q), the support of the curve s / → (x(q, s), y(q, s)). (See Figure 2.) Set
l=
0 1 L , y
where [·] denotes the integer part. For every k = 1, . . . , l, define % π $ qk = e 2 g e(k−1)yf (q0 ) ,
(4.20)
(4.21)
and, correspondingly, Qk = QL,d (qk ), σk (·) = σ (qk , ·), [−T1,k , T2,k ] = [−T1 (qk ), T2 (qk )], Wk = W (qk ) and 1k = 1(qk ). Let, moreover, 10 be the segment [−L, L] × {0} contained in W1 . For k ∈ {1, . . . , l − 1}, for every s ∈ (−L, L) ∩ (−T1,k , T2,k ), we want to identify the points (xk (s), yk (s)) ∈ 1k ⊂ Wk and (s, 0) ∈ Wk+1 . By construction, there exist a neighborhood U1k
580
Yacine Chitour and Mario Sigalotti
FIGURE 2
of (xk (s), yk (s)) in Qk , a neighborhood U2k of (s, 0) in Qk+1 , and an isometry ιk : U1k → U2k (with respect to the metric induced by M) such that ιk (U1k ∩ ∂Wk ) = U2k ∩ ∂Wk+1 . Consider the Riemannian two-dimensional manifold with boundary D(q0 ), obtained from the abstract union ∪1≤k≤l Wk by identification, for every k ∈ {1, . . . , l − 1} and every s ∈ [−L, L] ∩ [−T1,k , T2,k ], of (xk (s), yk (s)) ∈ Wk with (s, 0) ∈ Wk+1 . Since the gluing of two adjacent strips is rendered isometric (by the isometries ιk ), then the Riemannian structures induced by M on each of the strips Wk actually define a global Riemannian structure on D(q0 ). Let us assume, from now on, that
"
D(q0 )
K dA ≤ δ .
(4.22)
4.2. The fundamental tessellation Our purpose is to define a tessellation on D(q0 ), that is, a subdivision by geodesic polygons, in a checked pattern. Any such subdivision can be seen as a discrete system of coordinates globally defined on D(q0 ), assigning to any point the polygon which contains it.
A tessellation is determined by a grid of vertical and horizontal lines. The candidate vertical lines are the curves 1k defined above. Next lemma provides the estimates which allow to complete the desired construction.
Lemma 4.4. There exist δ0% ∈ (0, δ0 ), depending on K∞ , ε, and r, and two constants C(K∞ )
and C(ε, r) > 0 such that the following holds: Let q ∈ N verifies (4.16) with δ ≤ δ0% . Take t0 ∈ R, θ0 ∈ S 1 such that |t0 | ≤ L − y2 and |θ0 + π2 | ≤ δ. Define z0 = (t0 , 0, −θ0 ). Let (x(·), y(·), θ (·)) be the coordinates of σz0 (·) and denote by [0, T ] its maximal interval of definition in U W (q). Then (x(T ), & y(T )) ∈& 1(q) and |T − y| ≤ C(ε, r)δ. Moreover, for every s ∈ [0, T ], |x(s)| + |y(s) + s| + &θ (s) + π2 & ≤ C(K∞ )δ.
Proof of Lemma 4.4. every (x, y) ∈ W (q),
As it was done for estimates (3.5) and (3.6) of Section 3, we have, for 1 ≤ B(x, y) ≤ 1, 2
2 |F (x, y)| ≤ 2 K∞ .
Let I0 = [0, T0 ) be the largest interval (in [0, T ]) so that |θ (s) + π2 | < π3 for every s ∈ I0 . Since θ (0) = −θ0 , then T0 > 0. On I0 , the function v(s) = θ (s) + π/2 verifies the differential inequality 2 |v| ˙ ≤ 2 K∞ |v| , (4.23)
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
581
which implies that & √ π && & &θ (s) + & ≤ δe2 K∞ s . 2
Since y˙ = − cos v and |x| ˙ ≤ 2| sin v|, we have, for every s ∈ I0 , " s √ v(τ )2 dτ ≤ δ 2 se4 K∞ s , |y(s) + s| ≤ 0
(4.24)
(4.25)
and |x(s)| ≤ 2δse2
√ K∞ s
.
(4.26)
Recall that, due to Lemma 4.3, the coordinates (x, y) of a point in 1(q) satisfy |y −y| ≤ 2Lδ. It follows from (4.24)–(4.26) that, if δ is small enough with respect to d, then T0 = T and σ (T ) ∈ 1(q). Moreover, |T − y| ≤ C(ε, r)δ. The estimates on x(·), y(·), and θ(·) follow. For every j ∈ {−l +1, . . . , l −1}, let 3j be the support of the geodesic in D(q0 ) which starts at (j y, 0) ∈ 10 ⊂ W1 , making an oriented angle −π/2 with 10 . Assume that (4.22) holds, with δ ≤ δ0% . For every j ∈ {−l + 1, . . . , l − 1}, a repeated application of Lemma 4.4 to 3j and the successive 1k shows that, for every k ∈ {0, . . . , l}, 3j intersects 1k transversally, provided that δ is small enough with respect to y. Indeed, at every step of the+iteration, the angle determined by 3j and 1k at their point of intersection differs from π/2 by Dj,k K dA ≤ δ, where Dj,k is the geodesic quadrilateral of D(q0 ) bounded by 30 , 3j , 10 , and 1k . For every j ∈ {−l +1, . . . , l −1} and k ∈ {0, . . . , l}, denote by zj,k the point of intersection of 3j and 1k . Due to Lemma 4.4, the length of the portion of 3j connecting zj,k with zj,k+1 differs from y by at most C(ε, r)δ. For the same reason, the length of the portion of 1k which joins zj,k and zj +1,k differs from y by at most C(ε, r)δ. Denote by Pj,k the geodesic quadrilateral with vertices zj,k , zj,k+1 , zj +1,k+1 , and zj +1,k , for (j, k) ∈ {−l + 1, . . . , l − 2} × {0, . . . , l − 1}. The edges of Pj,k are portions of the horizontal and vertical lines 3j , 3j +1 , 1k , and 1k+1 . The family of all such Pj,k is called a tessellation on D(q0 ).
Correspondingly, a tessellation on the Euclidean plane, which covers the rectangle [0, ly] × [−(l − 1)y, (l − 1)y], is given by the family of squares Cj,k = [j y, (j + 1)y] × [ky, (k + 1)y] ,
(j, k) ∈ {−l + 1, . . . , l − 2} × {0, . . . , l − 1} .
Define T as the union of all ∂Cj,k ⊂ R2 , that is, 3 4 3 4 l T = ∪l−1 j =−l+1 [0, ly] × {j y} ∪ ∪k=0 {ky} × [−(l − 1)y, (l − 1)y] ,
and, similarly, T % as the union of all ∂Pj,k ⊂ D(q0 ). It follows from the above considerations that there exists a homeomorphism 4 : T −→ T %
% , whose restriction to any edge of any square C such that 4(zj,k ) = zj,k j,k is (1 + C(K∞ , ε, r)δ)Lipschitz continuous.
Consider a teardrop of size εr , starting from the point (0, 0) ∈ R2 in the direction (1, 0), contained in the Euclidean rectangle [0, (l +1)y]×[−(l −1)y, (l −1)y]. Such teardrop trajectory
582
Yacine Chitour and Mario Sigalotti
FIGURE 3
meets T in a sequence of points pm = (xm , ym ), 0 ≤ m ≤ V , numbered according to the order of intersection. (See Figure 3.) Notice that p0 = pV = (0, 0) .
(4.27)
The idea is now to define the correspondents of the points pm on T % , to equip them with a direction, and to construct a control strategy steering q0 to q0− , passing through all these intermediate states.
4.3. Reduction to the elementary problem In the following discussion, it simplifies the presentation to assume that r/ε, the size of the teardrop, is large with respect to y, the tessellation step. If this is the case, then each portion of teardrop which is contained in a square of the tessellation, considered as a planar curve, presents a small variation in its angular component. To this extent, we ask ε to be small with respect to y, r being by hypothesis greater than one. Such requirement is not restrictive, because y, according to its definition (4.18), depends only on K∞ , and our final goal is to prove the complete controllability of (Dε ) for every positive ε. In detail, fix ω > 0 and assume that, for every r > 1, the teardrop of size r/ε intersects the grid of step y in a sequence of points pm , 0 ≤ m ≤ V (V depending on r), such that (a)
the total variation of the angle component of the portion of teardrop connecting pm with pm+1 is smaller than ω, for every 0 ≤ m < V .
Assume that ω > 0 is small enough, so that, from every sequence pm as above, we can extract a subsequence, still denoted by pm , 0 ≤ m ≤ V , verifying (b) (4.27) holds; (c) the Euclidean distance between pm and pm+1 is larger than y/2, for every 0 ≤ m < V ; (d) the portion of teardrop connecting pm with pm+1 intersects T at most at one more point, for every 0 ≤ m < V . Notice that (d) implies that
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
(a’)
583
the total variation of the angle component of the portion of teardrop connecting pm with pm+1 is smaller than 2ω, for every 0 ≤ m < V .
An sequence in T % is associated with {pm }Vm=0 by % pm = 4(pm ) .
Consider the broken geodesic in R2 obtained by connecting, for every 0 ≤ m < V , pm to pm+1 with a segment Sm . Such curve is characterized by the lengths |Sm | of the segments Sm and −1 by the family {αm }Vm=0 , where αm is the oriented angle between pm+1 − pm and pm+2 − pm+1 . According to (c) and (d), we have 0 1 y √ |Sm | ∈ , 5y , (4.28) 2 while an upper bound on αm follows from √ the remark that |αm | is maximal when Sm and Sm+1 are two concatenated cords of length 5y of a circle of radius r/ε. (See Figure 4.) By easy trigonometric considerations, 5√ 6 5 ε |αm | ≤ arcsin εy ≤ C(K∞ ) . (4.29) r r
FIGURE 4
A first approximation of the desired teardrop in D(q0 ) is obtained by constructing a broken % . geodesic connecting all points pm −1 % Lemma 4.5. There exists a family {Sm% }Vm=0 of geodesic segments in D(q0 ) such that Sm
% % and p % % % % connects pm m+1 , ||Sm | − |Sm || ≤ C(K∞ , ε, r)δ, and the angle αm between Sm and Sm+1 ε % satisfies |αm | ≤ C(K∞ ) r .
Proof of Lemma 4.5.
A principal coordinate strip in D(q0 ) is a portion of the domain D(q0 ) of the following type: Either it is bounded by two curves 3j and 3j +4 (and we denote it by
584
Yacine Chitour and Mario Sigalotti
6j ) or by two curves 1k and 1k+4 (we write 6 k ). In 6j (respectively, 6 k ) we can consider the geodesic coordinates with base line 3j +2 (respectively 1k+2 ). Let θm ∈ S 1 be defined by
eiθm =
pm+1 − pm . |pm+1 − pm |
% and p % Assume that −θ < θm ≤ θ or π − θ < θm ≤ π + θ and take j such that pm m+1 belong to π π 3π 3π 6j . (When 2 − θ < θm ≤ 2 + θ or 2 − θ < θm ≤ 2 + θ the same procedure can be carried % % % , y % ), (x % % out on a strip of the type 6 k .) Let (xm m m+1 , ym+1 ) be the coordinates of pm , pm+1 in 6j . It follows from Lemma 4.3, Lemma 4.4, and the Lipschitz continuity of 4 that &$ % & &$ % & % % % % & x & & & m+1 − xm − (xm+1 − xm ) + ym+1 − ym − (ym+1 − ym ) ≤ C(K∞ , ε, r)δ .
Therefore,
and so
& & % & y% & & m+1 − ym & − tan θ & % m & ≤ C(K∞ , ε, r)δ , % & xm+1 − xm &
& & 5 6 % % & & ym+1 − ym & & − θ &arctan m & ≤ C(K∞ , ε, r)δ . % % & & xm+1 − xm
% Lemma 4.3 can be used to estimate the coordinate behavior of geodesics starting from pm or, to be more precise, of solutions of the system (4.6)–(4.8) with initial conditions of the type % , y % , θ % ), with |θ % | far from π/2. By standard continuity considerations, there exists a geodesic (xm m % joining p % and p % % % % segment Sm m m+1 , whose initial direction is given by (xm , ym , θm ), with & & % &θ − θm & ≤ C(K∞ , ε, r)δ (4.30) m
and
&& % & & &&S & − |Sm |& ≤ C(K∞ , ε, r)δ . m
(4.31)
θm%
Moreover, we can assume that is the unique initial condition such that √ (4.31) holds. Indeed, geodesic segments satisfying (4.31) have no conjugate points, since π/ K ∞ is a uniform lower bound on the value of conjugate times. Therefore, Sm is defined independently of the choice of the principal coordinate strip. % between S % and S % In order to estimate the angle αm m m+1 , we notice that it is not restrictive % % % to assume that −θ < θm , θm+1 ≤ θ and that pm , pm+1 , and pm+2 belong to the same principal % % , such that coordinate strip 6j . Hence, there exists an angle θm+1 , which characterizes Sm+1 % |θm+1 − θm+1 | ≤ C(K∞ , ε, r)δ.
% is evaluated by θ % !% !% In the coordinate system of 6j , αm m+1 −θm , where θm is the angle coordinate % % θm% | ≤ δ. of the tangent vector to Sm at pm+1 . Notice that, according to Lemma 4.3, |θm% − ! The relation between the measure of an angle in coordinates and its Riemannian value has been discussed in Remark 2.1, where it was noticed that the conversion mapping is Lipschitz continuous, with Lipschitz constant depending continuously on the value of B. Since B is, by hypothesis, uniformly separated from zero, it follows that there exists a universal constant C > 0 such that & %& & & &α & ≤ C &θ % −! θ% & m
m+1
≤
≤
m
C|αm | + C(K∞ , ε, r)δ ε C(K∞ ) , r
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
585
where the last equality follows from (4.29). Let βm be the oriented angle determined by the teardrop passing through pm and the segment Sm (with the agreement that the teardrop is oriented in its running sense and the segment from pm % ∈ U % D(q ) the unit vector which makes an angle β with S % . (When to pm+1 ). Denote by vm 0 m pm m % m = V , let vV be defined by (pV% , vV% ) = q0− .) In order to prove the existence of a trajectory in U D(q0 ) which connects q0 and q0− , admissible for the lifted Dubins problem, we will show % % % , v % ). The advantage is that all such , vm+1 ) is attainable from (pm that, for every m, (pm+1 m controllability subproblems, which we will call elementary problems, are essentially equivalent and that each of them “lives” in a single coordinate chart.
4.4. Solution of the elementary problem The m-th elementary problem is conveniently formulated in the geodesic coordinates of % . In such a coordinate system we have p % = (0, 0) and the rectangle whose base curve is Sm m % % % , the pm+1 = (|Sm |, 0). We noticed in Remark 2.1 that, at the points of the base curve Sm coordinate angle measures the true Riemannian angle between the corresponding unitary vector % . Therefore, what has to be solved is the motion planning problem defined by (2.8)–(2.10) and Sm % |, 0, α % + β with initial condition (0, 0, βm ) and final condition (|Sm m+1 ). m Fix a constant C0 = C(K∞ ) such that, & %& &α &, |βm | ≤ C0 ε m r
(4.32)
for every m.
In the Euclidean plane, the portion of teardrop connecting pm to pm+1 is a trajectory of x˙ = cos θ , y˙ = sin θ , u ∈ [−ε, ε] . (4.33) θ˙ = u , (x, y, θ )(0) = (0, 0, βm ),
. √ / Recall that |Sm | ∈ y2 , 5y . If ε is small enough with respect to d, then every solution of (4.33) intersects the surface , (x, y, θ ) ∈ R2 × S 1 | x = |Sm |
in a time close to |Sm |. Fix T > 0 such that every admissible trajectory of (4.33) intersects {x = |Sm |} only once, transversally, within time T . Let E(·) be the map which associates with an admissible control u : [0, T ] → [−ε, ε] the coordinates (y, θ ) of the trajectory corresponding to u(·), evaluated at the point of intersection with {x = |Sm |}. Notice that E is a $continuous % map from the space of admissible controls, endowed with the L1 topology, into R × − π2 , π2 .
The family of bang-bang control functions which are given by the concatenation of two arcs, the first one corresponding to control +ε and the second one to control −ε, form a continuous curve in L1 ([0, T ]), joining the two constant control functions u ≡ ε and u ≡ −ε. Taking into account also the two-bang concatenations where the controls are applied in the reversed order, it is clear that the family of all bang-bang control functions with at most two arcs forms a closed curve in the space of admissible controls. Choose a parameterization of such curve of$ the type % {us : [0, T ] → [−ε, ε]}s∈S 1 . Then γ : s / → E(us ) is a continuous closed curve in R × − π2 , π2 , which can be computed explicitly. In particular, straightforward computations show that there
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exists ρ = ρ(K∞ ) > 0 such that γ encircles the ball of center E(0) = (tan(βm )|Sm |, βm ) and radius ρε. Fix r = r(K∞ ) > 1 such that C0 ρ ≤ . r 6 Such a choice of r implies that γ encircles the segment & ' ε( & 6 = (0, θ) & |θ| ≤ 2C0 . r
Moreover, the distance from the support of γ to 6 is bounded from below by ρε/2, for δ small. Consider now the nonflat elementary problem θ x˙ = cos B , y˙ = sin θ, ˙ θ = u + F cos θ , (x, y, θ )(0) = (0, 0, β ) , m
u ∈ [−ε, ε] .
(4.34)
Fix any admissible control u : [0, T ] → [−ε, ε]. Denote by (x(·), y(·), θ (·)) (respectively, (x % (·), y % (·), θ % (·))) the solution of (4.33) [respectively, of (4.34)] corresponding to u(·). The same computations as in Lemma 4.3 imply that & & & & & & &x(t) − x % (t)& + &y(t) − y % (t)& + &θ (t) − θ % (t)& ≤ C(ε, K∞ )δ .
% In particular, we can assume that (x % (·), y % (·), θ % (·)), defined on [0, T ], intersects % |Sm |} once, $ π {xπ = % % % transversally. Define E (u) as the pair of coordinates (y (·), θ (·)) ∈ R × − 2 , 2 evaluated at the point of intersection. The map E % verifies & & &E(u) − E % (u)& ≤ C(ε, K∞ )δ .
Thus, the curve γ % : s / → E % (us ), which is closed and continuous, encircles 6, at least for δ small with respect to ε and 1/K∞ . By standard degree theory considerations, the image via E % of the space of admissible controls contains 6. Hence, the elementary problem is solvable, i.e., % % % , v % ) for the Dubins problem lifted on D(q ). It follows that (pm+1 , vm+1 ) is attainable from (pm 0 m q0− ∈ Aq0 and Theorem 4.1 is finally proved.
Acknowledgments The authors want to thank P. Pansu and K. Tapp for bringing to their attention interesting and useful results.
References [1]
Boscain U. and Chitour Y. On the minimum time problem for driftless left-invariant control systems on SO(3), Commun. Pure Appl. Anal. 1, 285–312, (2002).
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Boscain U. and Chitour Y. Time optimal synthesis for left-invariant control systems on SO(3), SIAM J. Control Optim., to appear.
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Busemann H. The Geometry of Geodesics, Academic Press, New York, (1955). Cheeger J. and Ebin D. Comparison Theorems in Riemannian Geometry, North Holland Publishing Co., (1975).
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Cohn-Vossen S. Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math. 2, 69–133, (1935). Croke C. and Karcher H. Volumes of small balls on open manifolds: Lower bounds and examples, Trans. Amer. Math. Soc. 309, 753–762, (1988). Dubins L. E. On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents, Amer. J. Math. 79, 497–516, (1957).
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Huber A. On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32, 13–72, (1957).
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Jurdjevic V. Geometric Control Theory, Cambridge Studies in Adv. Math. Cambridge University Press, (1997). Klingenberg W. A Course in Differential Geometry, Springer-Verlag, (1978).
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Krener A. A generalization of Chow’s theorem and the Bang-Bang theorem to non-linear control problems, SIAM J. Control 12, 43–51, (1974). Lobry C. Contrôlabilité des systèmes non linéaires, SIAM J. Control Optim. 8, 573–605, (1970).
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[14] Lobry C. Controllability of nonlinear systems on compact manifolds, SIAM J. Control. Optim. 12, 1–4, (1974). [15] Mittenhuber D. Dubins’ problem in the hyperbolic plane using the open disc model, geometric control and nonholonomic mechanics, (Mexico City, 1996), 115–152, CMS Conf. Proc. 25, Amer. Math. Soc., Providence, RI, (1998). [16] Mittenhuber D. Dubins’ problem is intrinsically three-dimensional, ESAIM Control Optim. Calc. Var. 3, 1–22, (1998). [17] Monroy-Perez F. Three-dimensional non-Euclidean Dubins’ problem, Geometric control and non-holonomic mechanics, (Mexico City, 1996), 153–181, CMS Conf. Proc. 25, Amer. Math. Soc., Providence, RI, (1998). [18]
Murray R. N., Li Z. X., and Sastry S. S. A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, (1994).
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Sharafutdinov V. A. The radius of injectivity of a complete open manifold of nonnegative curvature, Dokl. Akad. Nauk SSSR 231, 46–48, (1976). [22] Sigalotti M. and Chitour Y. Dubins’ problem on surfaces, II. Nonpositive curvature, SIAM J. Control Optim., to appear. [23] Sussmann H. J. Shortest three-dimensional paths with a prescribed curvature bound, Proc. IEEE CDC New Orleans, La, 3306–3312, (1995). [24]
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Received August 12, 2005 Laboratoire des signaux et systèmes, Université de Paris-Sud, CNRS, Supélec, Gif-sur-Yvette, France INRIA, Sophia-Antipolis, France
Communicated by Robert Greene
Dubins’ Problem on Surfaces. I. Nonnegative Curvature By Yacine Chitour and Mario Sigalotti
ABSTRACT. Let M be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any (p1 , v1 ) and (p2 , v2 ) in T M, is it possible to connect p1 to p2 by a curve γ in M with arbitrary small geodesic curvature such that, for i = 1, 2, γ˙ is equal to vi at pi ? In this article, we bring a positive answer to the question if M verifies one of the following three conditions: (a) M is compact, (b) M is asymptotically flat, and (c) M has bounded nonnegative curvature outside a compact subset.
1. Introduction Let (M, m) be a connected, oriented, complete Riemannian manifold and N = U M its unit tangent bundle. Points of N are pairs (p, v), where p ∈ M and v ∈ Tp M, m(v, v) = 1. Given ε > 0, Dubins’ problem consists of finding, for every (p1 , v1 ), (p2 , v2 ) ∈ N , a curve γ : [0, T ] → M, T ≥ 0, parameterized by arc-length such that γ (0) = p1 , γ˙ (0) = v1 , γ (T ) = p2 , γ˙ (T ) = v2 , with geodesic curvature bounded by ε and T as small as possible (depending on (p1 , v1 ), (p2 , v2 )). When the dimension of M is equal to two, Dubins’ problem can be formulated as the time optimal control problem for the following control system, (Dε ) :
q˙ = f (q) + ug(q) , q ∈ N, u ∈ [−ε, ε] ,
where f is the geodesic spray on N (i.e., f is the infinitesimal generator of the geodesic flow on M), g is the smooth vector field generating the fiberwise rotation with angular velocity equal to one and the admissible controls are measurable functions u : J → [−ε, ε], where J is an interval of R. The trajectories of (Dε ) are absolutely continuous curves γ = γu,q (·), with γ the solution of (Dε ) starting at q and associated with the admissible control u. A trajectory γ : [0, T ] → N of (Dε ) is said to be time optimal if, for every trajectory γ % : [0, T % ] → N of (Dε ) such that γ % (0) = γ (0) and γ % (T % ) = γ (T ), we have T ≤ T % . Note that, in the statement of Dubins’ problem, the existence of a curve γ of minimal length is not guaranteed. In the language of control theory, a controllability issue should be solved in
Math Subject Classifications. 14H55, 53C21, 93B05, 93B27. Key Words and Phrases. Controllability, geodesic curvature, nonnegative curvature. Acknowledgements and Notes. This work was partially supported through a European Community Marie Curie Fellowship in the framework of the Control Training Site (CTS), HPMT-CT-2001-00278. 2005 The Journal of Geometric Analysis ISSN 1050-6926
©
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order to tackle the time optimal problem. Recall that (Dε ) is completely controllable (CC) if, for every q1 , q2 ∈ N , q2 is reachable from q1 , i.e., there exists a trajectory of (Dε ) steering q1 to q2 . For q ∈ N, let Aq ⊂ N be the set of points of N reachable from q.
If M is the Euclidean plane, the dynamics defined by (Dε ) represents, in the robotics literature (cf. [18]), the motion of a unicycle (or a rolling penny) and the projections of the trajectories of (Dε ) on the plane are planar curves parameterized by arc-length with curvature bounded by ε. It is easy to see that (Dε ) is completely controllable for every ε > 0 and, given any pair (p1 , v1 ), (p2 , v2 ) ∈ U R2 , there exists a time optimal trajectory of (Dε ) connecting (p1 , v1 ) and (p2 , v2 ). In 1957, Dubins [8] determined the global structure of time optimal trajectories of (Dε ) in the case where M is the Euclidean plane: He showed that such trajectories are concatenations of at most three pieces made of circles of radius 1ε or straight lines. Further restrictions on the length of the arcs of an optimal concatenation have been proved by Sussmann and Tang [24].
Dubins’-like problems have been proposed by considering more general manifolds M. For instance, the case where M is a two-dimensional manifold of constant Gaussian curvature was investigated in [2, 5, 10, 15] and the case where M = Rn , S n , n ≥ 3 was studied in [16, 17, 23]. Another line of generalization consists of considering the distributional version of (Dε ). For simplicity, suppose M to be two-dimensional. The distributional dynamics can be represented by the two-input control system (DDε ) : q˙ = uf (q) + vg(q) with |u|, |v| ≤ ε (cf. [1]). The controllability issue is trivial since it can be solved infinitesimally: Let h = [f, g], where [., .] denotes the Lie bracket; then the distribution (f, g) is strongly bracket generating, i.e., for every q ∈ N , the triple (f (q), g(q), h(q)) spans Tq N .
In this article, we follow the first path of generalization, i.e., we assume that M is a twodimensional connected Riemannian manifold, oriented and complete (with possibly nonconstant curvature). Our aim is to find geometric or topological conditions on M, such that, for every ε > 0, (Dε ) is completely controllable. We refer to that property as the unrestricted complete controllability (UCC) for Dubins’ problem (we still use the word “Dubins” although we will not consider any optimal control problem). Geometrically, the (UCC) property can be stated as follows: For every (p1 , v1 ), (p2 , v2 ) ∈ N , there exists a curve γ connecting p1 to p2 with prescribed initial and final directions v1 and v2 and with arbitrary small geodesic curvature.
To establish (CC) of (Dε ), ε > 0, we use a standard reduction (cf., for instance, [10]): We will show that (Dε ) is completely controllable if and only if (Dε ) is weakly symmetric, i.e., for every q = (p, v) ∈ N , q − = (p, −v) ∈ Aq . If, for instance, M = R2 , then a control strategy which shows that (Dε ) is weakly symmetric can be given by u so that the resulting trajectory is a teardrop of size 1ε . For R > 0 a teardrop of size R is described in Figure 1. ! then #∗ |N ! → M be a Riemannian covering and (D !ε ) be Dubins’ problem on M; Let # : M ! maps trajectories of (Dε ) onto trajectories of (Dε ). Therefore, if the (CC) property holds for !ε ), then it also holds for (Dε ). Equivalently, if (D !ε ) is weakly symmetric, then (Dε ) is. For (D instance, if M is flat, then a controllability strategy for M is obtained by projecting the one of the Euclidean plane, seen as the universal covering of M. This simple idea of applying strategies which are valid on a Riemannian covering of the manifold (not necessarily a universal covering) will be repeatedly exploited in the article. The first condition ensuring (UCC) which we obtain is purely topological: If M is compact, then, by means of a Poisson stability argument, (UCC) turns out to hold for Dubins’ problem. Thus, we are let to the case where M is noncompact. The geometric quantity which plays a crucial role in the characterization of controllable Dubins’ problems is the Gaussian curvature of M, denoted by K. The curvature appears quite soon in the study of the Lie bracket configuration defined by (Dε ) and, therefore, of local controllability
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
FIGURE 1
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The teardrop trajectory of size R.
issues: Indeed, for every q ∈ N , [f, [f, g]](q) = −K(π(q))g(q) ,
(1.1)
where π : N → M denotes the bundle projection. The role of K is reinforcedly suggested by the following fact: If M is the Poincaré half-plane (K ≡ −1), then (Dε ) is completely controllable if and only if ε > 1 (cf. [5, 15]). Roughly speaking, this happens because the negativeness of K not only prevents the geodesics to have conjugate points but actually is an obstacle for the controlled turning action to overcome the spreading of the geodesics. When |u| ≤ ε ≤ 1, the steering effect of u g is not strong enough and (CC) fails to hold. It is therefore natural to formulate necessary conditions for (UCC) in terms of the Gaussian curvature K. For instance, an extension of the case K ≡ 0 is given by the situation in which M is asymptotically flat, i.e., K tends to zero at infinity. Under this hypothesis, we are able to prove the (UCC) property: The control strategy is based on the possibility of tracking a teardrop loop in a covering domain over a piece of M at infinity. We will see that a suitable covering manifold can be globally described by a single appropriate geodesic chart. Bearing in mind the previous example of noncontrollability, it is reasonable to study first the situation where the curvature is nonnegative. The negative curvature case is the subject of subsequent article [22], where we provide a complete characterization of two-dimensional nonpositively curved manifolds M, with either uniformly negative or bounded curvature, that satisfy property (UCC). Such characterization involves the limit set of M as well as an integral decay condition of K at infinity. In the nonnegative curvature case, the one treated here, no local spreading effect due to the drift term has to be compensated. A result by Cohn-Vossen (cf. [6]) implies that, if K ≥ 0 and K is not identically equal to 0, then M is diffeomorphic to a plane and, more importantly for the
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controllability issue, "
M
K dA ≤ 2π ,
(1.2)
where dA is the surface element in M. (It is actually true that, in any dimension, a complete noncompact Riemannian manifold whose sectional curvatures are all everywhere nonnegative and with one point where they all are positive is diffeomorphic to an Euclidean space, as proved by Perelman in [19].) As a consequence of (1.2), for any fixed radius R > 0, the total curvature on the disk centered at p with radius R tends to zero as p tends to infinity. The same is true under the relaxed hypothesis that K is nonnegative outside a compact subset of M. The integral decay of K to zero can be interpreted as a kind of asymptotic flatness condition and it suggests that (Dε ) should be completely controllable for every ε > 0. We are able to confirm that intuition under the additional assumption that K is bounded over M, i.e., K∞ = supM K is finite. The existence of a control strategy which allows to track a teardrop loop at infinity is much more delicate to prove than in the asymptotically flat case. The key step is the identification of a simply connected covering domain on which the teardrop strategy can be applied. The covering domain D is not described anymore by one single chart, as in the asymptotic flat case, but by gluing rectangular strips, each of them obtained by one regular geodesic chart. There are O( 1ε ) such strips and each of them has width proportional to √K1 . Then, using these strips, one is able to mesh D by ∞
geodesic quadrilaterals Pj,k with edges of length proportional to
√1 . K∞
The tracking operation is
now decomposed in O( 1ε ) steps: We design a discrete approximation of the teardrop, by fixing a sequence of O( 1ε ) points on the edges of the polygons Pj,k and by associating with each of them a corresponding direction. After that, we solve the problem of connecting pairs of subsequent points of the approximating sequence by an admissible trajectory, being tangent to the associated directions. Each elementary problem of this kind can be formulated in a single coordinate strip. Intuitively, its solution is based on the topological description of small time attainable sets for nondegenerate single-input control-affine three-dimensional systems, due to Lobry [13]: The set of points which are reachable from q0 ∈ N in small time, is given by the region enclosed by two surfaces, obtained as union of all small-time bang-bang trajectories from q0 with one switch. What we do in practice, is to estimate the coordinate expression of such surfaces and to check whether they enclose the final state of the elementary problem. The article is organized as follows. In Section 2, we gather the notations used in the article, describe the general construction of local covering domains, establish basic properties for Dubins’ problem and, finally, study the case where M is compact. Section 3 is devoted to the asymptotically flat case, where the Gaussian curvature tends to zero at infinity. In Section 4, the unrestricted complete controllability property is established when K is nonnegative and bounded outside a compact domain.
2. Basic notations and first results 2.1. Differential geometric notions Let (M, m) be a complete, connected, oriented, two-dimensional Riemannian manifold. Denote by K its Gaussian curvature and by N the unit tangent bundle U M. Let π : N → M be the canonical bundle projection of N onto M. We will usually denote by p a point in M and by q = (p, v) one in N, where p = π(q) and v ∈ Tp M, m(v, v) = 1. Given v ∈ Tp M, we write v ⊥ for its counterclockwise rotation in Tp M of angle π/2. For every q = (p, v) ∈ N , we set q ⊥ = (p, v ⊥ ) and q − = (p, −v).
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Given p1 , p2 ∈ M, d(p1 , p2 ) denotes the geodesic distance between p1 and p2 . When no confusion is possible, we simply write -p- (respectively, -q-) to denote the distance d(p, p0 ) (respectively, d(π(q), p0 )) from a fixed point p0 ∈ M.
Let f be the geodesic spray on T M, whose restriction to N (still denoted by f ) is a welldefined vector field on N. Recall that f is characterized by the following property: p(·) is a geodesic on M if and only if (p(·), p(·)) ˙ is an integral curve of f . In particular, f satisfies the relation π% (f (q)) = q .
(2.1)
Denote by g the smooth vector field on N, whose corresponding flow at time t is the fiberwise rotation of angle t. In terms of the covariant derivative on M, the integral curves of f and g are solutions, respectively, of the following equations, # # p˙ = 0 , p˙ = v , and ∇v v = 0 , ∇v v = v ⊥ . We write etf (respectively, etg ) to denote the flow of f (respectively, g) at time t. If x0 belongs to a metric space (X, d0 ) and ρ > 0, then Bρ (x0 ) denotes the open ball of center x0 and radius ρ. Given a subset Y of X, Clos(Y ) and Int(Y ) are, respectively, the closure and the interior of Y . In the sequel of the article, we will systematically use as local coordinates the geodesic ones, whose definition is recalled below. Its construction has a crucial role in the present exposition, since it allows to define a wide class of local covering domains of M. Given q ∈ N, consider the map R2 (x, y)
φq :
−→ /−→
M $ % π π eyf e 2 g exf (q) .
Fix R = [x1 , x2 ] × [y1 , y2 ] ⊂ R2 and assume that the origin (0, 0) belongs to R. If φq is a local diffeomorphism at every point of R, then R can be endowed with the Riemannian structure lifted from M, in such a way that φq becomes a local isometry. If this happens, we denote by R(q) the manifold with boundary which is obtained. The segment [x1 , x2 ] × {0}, which is the support of a geodesic in R(q), is called the base curve of R(q). The Gaussian curvature of R(q) at a point (x, y) is given by K(φq (x, y)), and, where no confusion can arise, will be denoted by & & K(x, y). If R is a neighborhood of (0, 0) and φq &R is injective, then φq &R is a geodesic chart on M. In the coordinates (x, y), m has the form m(x, y) = B 2 (x, y) dx 2 + dy 2 ,
where B : R → R is the solution of the system B(x, 0) ≡ 1,
By (x, 0) ≡ 0,
and
Byy + KB = 0 ,
(2.2)
in which the index y appearing in By , Byy stands for the partial differentiation with respect to y. (See, for instance, [11].) Notice that, for every point q ∈ N and every small enough rectangular neighborhood R & of (0, 0), φq &R is a geodesic chart on M. In general, if B is the solution of (2.2) on R, with
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K = K ◦ φq , then R(q) is well defined if and only if B is everywhere positive on R. In R(q), define the real-valued function F by F (x, y) =
By (x, y) . B(x, y)
(2.3)
The unit bundle U R(q) can be identified with & ( ' & (x, y, vx , vy ) ∈ R4 & (x, y) ∈ R, B 2 (x, y)vx2 + vy2 = 1 .
Equivalently, a set of coordinates in U R(q) is given by (x, y, θ ) ∈ R × S 1 , with the identification Bvx = cos θ,
vy = sin θ .
(2.4)
Remark 2.1. Notice that, for all points (x, y, θ ) in U R(q) such that y = 0, the coordinate θ
measures the Riemannian angle between the corresponding unit vector and the base curve of R(q). On the other hand, a unit vector of coordinates (x, y, 0) or (x, y, π ) is always φq∗ m-orthogonal to the segment {x} × [y1 , y2 ].
More generally, the function which associates with θ the angle between (x, y, 0) and (x, y, θ ) is a Lipschitz continuous map from S 1 into itself, with Lipschitz constant continuously depending on B(x, y). In geodesic coordinates, f and g are given by ) *T cos θ f (x, y, θ ) = , , sin θ, F (x, y) cos θ B(x, y)
g(x, y, θ ) = (0, 0, 1)T .
(2.5)
The pair of vector fields (f, g) define a contact distribution on N , i.e., the triple (f (q), g(q), [f, g](q)) spans Tq N for every q ∈ N, where [·, ·] stands for the Lie bracket. The Lie-algebraic structure of the contact distribution {f, g} is characterized by the relations (i) [f, g] = h ,
(ii) [g, h] = f ,
(iii) [h, f ] = Kg ,
where h, defined by (i), is represented in geodesic coordinates as *T ) sin θ , − cos θ, F (x, y) sin θ . h(x, y, θ ) = B(x, y)
(2.6)
(2.7)
A proof of (2.6) can be obtained, for instance, by using the expressions (2.5) of f and g in geodesic coordinates. Equivalently, (2.6) could have been derived from the structure equations arising from the moving frame approach (see [5]). A metric m ! on N can be introduced by requiring that (f (q), g(q), h(q)) is an m !-orthonormal basis of Tq N , for every q ∈ N . Such m ! is usually called the Sasaki metric inherited from m and endows N with a complete Riemannian structure. (See, for instance, [20].)
2.2. The control system Recall that, for every ε > 0, (Dε ) denotes the control system (Dε ) :
q˙ = f (q) + ug(q) ,
q ∈ N,
u ∈ [−ε, ε] .
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By definition, an admissible control is a measurable function u(·), defined on some interval of R, with values in [−ε, ε]. The solutions of (Dε ) corresponding to admissible controls are called admissible trajectories. It follows from (2.1) and the definition of g that, for every admissible trajectory q : [0, T ] → N of (Dε ), d(π(q(0)), π(q(T ))) ≤ T . Therefore, M being complete, for every admissible control u : R → [−ε, ε], the non-autonomous vector field f + u(t)g is complete, that is, with any initial condition q0 ∈ N we can associate a solution q(·) of (Dε ), defined on the whole real line, such that q(0) = q0 . In other words, the control system (Dε ) is complete. For every q ∈ N and T > 0, the attainable set from q within time T is the set ATq consisting of the endpoints of all admissible trajectories starting from q and defined on a time interval of length smaller than T . Similarly, the attainable set from q is the set Aq consisting of the endpoints of all admissible trajectories starting from q. The control system (Dε ) is called completely controllable if Aq = N for every q ∈ N.
Definition 2.2. We say that Dubins’ problem on M has the unrestricted complete controllability (UCC) property if, for every ε > 0, (Dε ) is completely controllable. In local geodesic coordinates, (Dε ) can be written as follows, x˙
y˙ θ˙
=
= =
cos θ , B sin θ , u + F cos θ .
More intrinsically, we can rewrite system (2.8)–(2.10) in the form # p˙ = v , ∇v v = uv ⊥ ,
(2.8) (2.9) (2.10)
(2.11)
which accounts for a clear geometric interpretation of the unrestricted controllability property: The Dubins problem on M is unrestrictedly completely controllable if and only if, for every (p1 , v1 ), (p2 , v2 ) ∈ N , for every ε > 0, there exists a curve p : [T1 , T2 ] → M with geodesic curvature smaller than ε such that p(Ti ) = pi , p(T ˙ i ) = vi , i = 1, 2.
The fact that f and g define a contact distribution on N has the important consequence that, for every 0 < t < T and q ∈ N, etf (q) belongs to Int(ATq ). This follows, for instance, from the description of small-time attainable sets for single-input nondegenerate three-dimensional control systems given by Lobry in [13]. From the viewpoint of control theory, the property that the distribution defining (Dε ) has a contact structure implies that (Dε ) is bracket generating, i.e., such that the iterated Lie brackets of f and g span the tangent space to N at every point.
Remark 2.3. If q : [0, T ] → N is a trajectory of (Dε ) corresponding to some admissible
control u : [0, T ] → [−ε, ε], then the trajectory q(T − ·)− obtained from q(·) by reflection and time-reversion is itself an admissible trajectory of (Dε ) and steers q(T )− to q(0)− . Its corresponding control function is given by −u(T − ·), which is indeed admissible. Therefore, for every q, q % ∈ N , q % belongs to Aq if and only if q − belongs to A(q % )− .
Remark 2.4. Assume that, for every q in N , q − ∈ Aq , i.e., that (Dε ) is weakly symmetric.
Then, due to Remark 2.3, q % ∈ Aq if and only if Aq = Aq % . It follows that, for every q ∈ N and
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every q % ∈ Aq ,
% $ % $ % $ q % ∈ Int Ae−tf (q % ) = Int Aq % = Int Aq ,
where t > 0 and the first inclusion follows from the previously quoted Lobry’s result. Therefore, {Aq }q∈N is an open partition of N . Since N is connected, then (Dε ) is completely controllable.
Thanks again to Remark 2.3, we obtain the following equivalence: (Dε ) is completely controllable if and only if, for every q ∈ N , there exists q % ∈ Aq such that (q % )− ∈ Aq % .
Remark 2.5. If M is nonorientable, then the vector field g is not well defined. Anyhow, the control problem still makes sense, since locally g can be defined fixing arbitrarily an orientation, and the system is independent of this choice. Formally, Dubins’ problem can be defined as a control-affine system with multiple controls, using a partition of unity on M in order to glue the local definitions of g together. It is known that M admits an oriented Riemannian double ! → M. Since the additional hypothesis under which we get the (UCC) property covering # : M (cf. Proposition 2.7 and Theorems 3.1, 4.1) are shared by any finite Riemannian covering, then the results of this article extend to nonorientable manifolds. A sufficient condition for unrestricted complete controllability is the compactness of M. This fact is a consequence of a more general result on controllability on compact manifolds of bracket generating systems made of conservative vector fields due to Lobry [14]. Lemma 2.6 gives a stronger formulation of Lobry’s result, adapted to the specific control system (Dε ), which implies also that every attainable set is unbounded when M is open. The proof is a variation on the classical one of Poincaré’s theorem on volume-preserving flows.
Lemma 2.6. If q ∈ N exists such that Aq is relatively compact in N, then M is compact and Aq = N.
Proof of Lemma 2.6.
Fix q ∈ N and assume that G = Clos(Aq ) is compact in N. As already remarked, for every t > 0 and every q % ∈ N, etf (q % ) ∈ Int(A2t q % ). Fix t = 1. The compactness of 2 % G and the continuous dependence of Aq % on q imply that there exists ρ > 0 such that, for every q % ∈ G, $ $ %% Bρ ef q % ⊂ Aq % .
(2.12)
We want to prove that ∂Aq is empty. Let, by contradiction, r ∈ ∂Aq . A well-known theorem by Krener [12] states that any attainable set of a bracket generating system is contained in the closure of its interior. Therefore, V = Aq ∩ Bρ (r) has nonempty interior and, in particular, its volume is strictly positive. Since ef is a volume preserving diffeomorphism of N (see, for instance, [20]) and Aq has finite volume (it is bounded), then {enf (V )}n∈N cannot be a disjoint family, being enf (V ) ⊂ Aq for every n ∈ N. Therefore, there exist n1 < n2 such that en1 f (V ) ∩ en2 f (V ) is not empty. Equivalently, there exists a point in e(n2 −n1 −1)f (V ) whose image by ef lies in V . Due to (2.12), it follows that r ∈ Int(Aq ) and the contradiction is reached.
Proposition 2.7. Let M be a complete, connected, oriented, two-dimensional Riemannian manifold. Assume, in addition, that M is compact. Then Dubins’ problem is unrestrictedly completely controllable. For the rest of the article, we deal with the case M noncompact.
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3. Asymptotically flat manifolds Throughout this section, we assume that M is asymptotically flat, that is, lim K(p) = 0 .
-p-→∞
(3.1)
For every L > 0, let QL = [0, 2L] × [−L, L]. According to the notation introduced in Section 2.1, if the map φq0 , q0 ∈ N , is a local diffeomorphism at every point of QL , then QL (q0 ) denotes the Riemannian manifold (with boundary) obtained endowing QL with the Riemannian structure lifted from M. Let us characterize values of L for which the construction of QL (q0 ) can be carried out. Let B be the solution of (2.2) on QL , with K = K ◦ φq0 . Set & & (3.2) δ = max &K ◦ φq0 & . QL
By Sturm-Liouville theory, we can compare B with the solution of (2.2) √ √ corresponding to K constantly equal to δ. We obtain that, if δ|y| ≤ π2 , then B(x, y) ≥ cos( δy) ≥ 0 for every x ∈ [0, 2L]. Thus, if π L< √ , 2 δ
(3.3)
then QL (q0 ) is well defined. In particular, since M is asymptotically flat, then, for every L > 0 and every q0 outside a compact subset of N (depending, in general, on L), QL (q0 ) is well defined. We stress that no global finiteness property is stated (nor needed) for the projection φq0 from QL (q0 ) onto its image. In general, for L fixed, the cardinality of the set of preimages φq−1 (q0 ) 0 can fail to have a uniform bound when q0 varies in N . The situation will be different in Section 4. Together with m, also the control problem (Dε ) is lifted from N to U QL (q0 ). Let us stress the trivial, but crucial, property that every admissible trajectory of the lifted control system is projected by φq0 to an admissible trajectory of (Dε ). In the coordinates (x, y, θ ) of U QL (q0 ), the dynamics of the lifted system is described by (2.8)–(2.10). Due to Remark 2.4 and Lemma 2.6, the proof of the complete controllability of (Dε ) reduces to show that q0− ∈ Aq0 if δ is small enough. This will be done by designing an admissible trajectory for the lifted control problem on U QL (q0 ), steering (0, 0, 0) to (0, 0, π). Fix q0 ∈ N , L > 0 and assume that
√ π δ≤ , 3L
(3.4)
where δ is defined as in (3.2). The Sturm-Liouville theory, together with the well definedness of QL (q0 ), implies that $√ % $√ % cos δy ≤ B(x, y) ≤ cosh δy , (3.5) and
$√ % & & & By (x, y) & √ sinh δ|y| & & |F (x, y)| = & ≤ δ $√ % , B(x, y) & cos δy
(3.6)
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for every (x, y) ∈ QL (q0 ). An upper bound for |F | in QL (q0 ) is given by we can assume that ε max |F | ≤ , QL (q0 ) 2
√
√
δ sinh(√ δL) . Then, cos( δL)
(3.7)
by taking √ δ≤
ε $ %. 4 sinh π3
(3.8)
Consider now the control system (Dε/2 ) on the unit bundle of the Euclidean plane. Let u(·) be the control function corresponding to the trajectory whose projection on R2 is a teardrop of size 2/ε which leaves the origin horizontally and arrives at the origin with the opposite direction (Figure 1, with R = 2/ε). Thus, u(·), is piecewise constant, taking alternately the values −ε/2 and ε/2. Denote the coordinates of the teardrop trajectory in R2 × S 1 by x(·), y(·) and θ(·). It √follows from straightforward computations that (x(·), y(·)) takes values in the rectangle [0, 2( 3 + 1)/ε] × [−2/ε, 2/ε] and that the teardrop has length 14π 3ε . Fix L=
3 . ε
The idea is to apply to the lifted system the time-variant feedback control u(t) = u(t) − F (x, y) cos θ ,
(3.9)
which is admissible, as long as the corresponding trajectory stays in U QL (q0 ), since (3.7) holds. Consider the solution γ (·) = (x(·), y(·), θ (·)) of (2.8)–(2.10) corresponding to u(·), with initial condition γ (0) = (x0 , 0, 0). As long as (x(t), y(t)) stays in QL , we have y(t) = y(t) and θ (t) = θ (t). Therefore, & & " t & & 1 |x(t) − x(t) − x0 | ≤ | cos(θ (s))| && − 1&& ds B(x(t), y(t)) 0 & & & & 14π 1 ≤ max && − 1&& . 3ε QL (q0 ) B $ % It follows from (3.5) that, for every α ∈ 0, π2 , if √ δL ≤ α , (3.10) then & & &1 & cosh(α) − 1 & max & − 1&& ≤ . QL (q0 ) B cos(α)
Therefore, it is possible to fix α, independent of ε, such that, whenever δ satisfies (3.10), & & &1 & 14π 1 max && − 1&& ≤ . 3ε QL (q0 ) B 4ε
1 Assume that (3.10) is satisfied and fix x0 = 4ε . Then γ (·) is defined for the entire time duration of u(·). At its final point, its coordinates are of the type (x1 , 0, π). Concatenating γ with two trajectories corresponding to control equal to zero, we obtain an admissible trajectory
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for Dubins’ problem lifted to U QL (q0 ), steering (0, 0, 0) to (0, 0, π). We proved the following theorem.
Theorem 3.1. Let M be a complete, connected, oriented, two-dimensional Riemannian manifold. Assume, in addition, that M is asymptotically flat. Then, Dubins’ problem is unrestrictedly completely controllable. Actually, from the nature of the above argument, a stronger result follows.
Proposition 3.2. There exists a universal constant µ > 0 such that, if lim sup-p-→∞ |K(p)| ≤ µε2 , then (Dε ) is completely controllable.
Proposition 3.2 can be recovered from the smallness conditions (3.4), (3.8), and (3.10) imposed on δ, where L should be replaced by 3/ε and α can be given explicitly.
4. Manifolds with nonnegative curvature outside a compact set 4.1. Construction of the covering domain From now on, in addition to the general assumptions on M + made in Section 2, we will assume K to be nonnegative outside a compact subset of M. Since M K dA, the total curvature of M, is well defined (allowing extended values), and larger than −∞, then it follows from a result by Huber [9], that M is finitely connected. Therefore, Cohn-Vossen theorem [6] applies, i.e., " K dA ≤ 2π χ (M) , (4.1) M
where χ (M) is the Euler characteristic of M. The main result of the section is the following.
Theorem 4.1. Let M be a complete, connected, oriented, two-dimensional Riemannian manifold. Assume , in addition, that the Gaussian curvature K of M is bounded and {p ∈ M| K(p) < 0} is relatively compact in M. Then, Dubins’ problem is unrestrictedly completely controllable. In the sequel, we assume that K is bounded on M and we set K∞ = sup K(p) . p∈M
Without loss of generality, K∞ > 0. than
For every L, d > 0, let QL,d = [−L, L]×[−d, d]. As remarked in Section 3, if d is smaller √π , then the local covering domain QL,d (q0 ) is well defined. Fix 2 K∞
π . d= √ 3 K∞
(4.2)
"
(4.3)
We claim that, for every L > 0, lim
-q-→∞ QL,d (q)
K dA = 0 .
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The result follows from general properties of Riemannian surfaces. Recall that the injectivity radius at a point p of M is defined as the least upper bound of all R > 0 such that the exponential map expp : Tp M −→ M , restricted to the disk BR (0), is injective. It is denoted by ip (M), while i(M) = inf p∈M ip (M) is called the injectivity radius of M. Property (4.3) is proved as soon as we show that i(M) > 0, since, due to Cohn-Vossen theorem, for every R > 0, lim
"
-p-→∞ BR (p)
K dA = 0 .
(4.4)
Lemma 4.2. Let M be a complete, connected, two-dimensional Riemannian manifold. Assume that K is bounded on M and nonnegative outside a compact subset. Then i(M) > 0. We were not able to find in the literature an explicit statement of Lemma 4.2, although it happens to be a rather straightforward consequence of known facts. It worth mentioning, anyhow, that, in the special case when 0 ≤ K ≤ K∞ , a positive lower bound on the injectivity radius of M is known. Indeed, a theorem √ by Sharafutdinov [21, 7] states that i(M) is larger than or equal to the minimum between π/ K ∞ and the injectivity radius of the Cheeger-Gromoll soul of M, which is a compact submanifold of M. In the nonflat two-dimensional case, the soul is always atomic and its injectivity radius can be set to be +∞.
Proof of Lemma 4.2.
A general result [4, Lemma 5.6], which holds for any complete Riemannian manifold, states that, for every p ∈ M, ip (M) = min{t > 0 | t is a conjugate time for a geodesic γ : [0, +∞) → M, γ (0) = p , or 2t is the length of a geodesic loop passing through p} . (4.5)
Fix a compact set M0 of M such that {p ∈ M| K(p) < 0} ⊂ M0 . We can assume that ClosM \ M0 is a finite union of tubes of M, that is, according to the definition of Busemann [3], subsets of M which are homeomorphic to half-cylinders and whose boundaries are simple closed geodesic polygons. Moreover, due to Cohn-Vossen theorem, we can suppose that the total curvature of each tube is smaller than π. Fix one of such tubes and denote it by T . We have to prove that inf p∈T ip (M) > 0. Let γ be a simple geodesic loop contained in the interior of T . Then γ identifies two connected regions of T , a bounded and an unbounded one. The bounded region B must contain the boundary of T , otherwise Gauss-Bonnet theorem would constrain the total curvature of B to be greater than or equal to π, contradicting the assumptions made on T . Property (44.16) in [3] (originally stated in the general framework of G-surfaces) implies that there exists a curve in T , freely homotopic to ∂T , of minimal length. In particular, the length of γ is bounded from below by a positive constant. On the other hand, since K ≤ √ K∞ , the first conjugate time for any half geodesic contained in M is bounded from below by π/ K∞ . The lemma follows from (4.5). Let us show how the integral smallness of K on the covering domains QL,d (q), ensured — at infinity — by (4.3), allows to estimate the evolution of geodesics in coordinates. Let p(·) be a geodesic in QL,d (q), parameterized by arc-length; then (p(·), p(·)) ˙ is a curve
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
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in U QL,d (q), whose coordinates satisfy x˙
=
y˙ θ˙
= =
cos θ , B sin θ , F cos θ ,
(4.6) (4.7) (4.8)
which is a particular case of (2.8)–(2.10). Denote by σz the curve in U QL,d (q), which is solution of the system (4.6)–(4.8) with initial condition σz (0) = z ∈ U QL,d (q), and let [−T1,z , T2,z ] be its maximal interval of definition. Let
, $ % N 0 = q ∈ N | d π(q), K −1 ((−∞, 0)) > L + d ,
and remark that the complement of N 0 in N is compact. The set N 0 is defined in such a way that, for every q0 ∈ N 0 , K is nonnegative on QL,d (q0 ). Fix
π 4
0 verify Q K dA ≤ δ ≤ δ0 , then, for every z0 = (0, y0 , θ0 ) with . / y0 ∈ [− d2 , d2 ] and θ0 ∈ − θ , θ , the corresponding σz0 (·) = (x(·), y(·), θ (·)) satisfies (i)
for every t in [−T1,z0 , T2,z0 ],
|x(t) − cos(θ0 )t|
|y(t) − y0 − sin(θ0 )t| |θ (t) − θ0 |
Moreover, if |θ0 | ≤ (ii)
d 8L ,
≤
≤ ≤
(2L + d)δ , 2Lδ , δ.
(4.9) (4.10) (4.11)
then
x(−T1,z0 ) = −L and x(T2,z0 ) = L.
Proof of Lemma 4.3. every (x, y) ∈ Q,
As it was done for (3.5) in Section 3, one obtains from (4.2) that, for 1 ≤ B(x, y) ≤ 1 . 2
(4.12)
Fix z0 = (0, y0 , θ0 ) and σz0 (·) = (x(·), y(·), θ (·)) as in the statement of the lemma. Let [−T1 , T2 ] = [−T1,z0 , T2,z0 ] and denote by (−t1 , t2 ) the maximal open neighborhood of zero (in [−T1 , T2 ]) such that, for every t ∈ (−t1 , t2 ), |θ (t)|
0, for every q outside a compact subset of N (depending, in general, on δ), " and K dA ≤ δ . (4.16) K ≥ 0 on QL,d (q) QL,d (q)
We can choose δ ≤ δ0 (defined as in Lemma 4.3), and fix q0 ∈ N such that (4.16) holds for every q verifying d(π(q), π(q0 )) ≤ 2L. We next build a Riemannian two-dimensional manifold, which will be a finite covering of a region of M containing π(q0 ). The covering domain will be obtained by gluing together several rectangles of the type QL,d (q). The purpose is to track a teardrop of size r/ε, r > 1 to be fixed, for the lifted Dubins problem and to eventually obtain that q0− ∈ Aq0 . In this perspective, since L will measure the size of the covering domain, we fix L=
4r . ε
(4.17)
The (UCC) property will follow, as in the asymptotic flat case, from the fact that the tracking operation can be performed for any q0 outside a big enough compact subset of N (depending, in general, on ε). From now on, we assume, sometimes implicitly, that δ is as small as needed with respect to ε, 1/r and 1/K∞ . Notice that r is considered independent of ε and K∞ . We will denote by C(·) any constant which is a function of the quantities appearing in its argument. For instance, C(K∞ , ε, r) denotes a constant which depends on K∞ , ε, and r. Fix y=
d . 8
(4.18)
For every q verifying (4.16), let σ (q, ·) = (x(q, ·), y(q, ·), θ (q, ·)) be the solution in U QL,d (q) of the system (4.6)–(4.8), with initial condition (x(q, 0), y(q, 0), θ (q, 0)) = (0, −y, 0) . Denote by [−T1 (q), T2 (q)] the maximal interval of definition of σ (q, ·). If δ is small enough, then we can assume that y(q, ·) takes only negative values. Let W (q) be the region of QL,d (q) defined by W (q) = {(x(q, s), t)| s ∈ [−T1 (q), T2 (q)], t ∈ [y(q, s), 0]} ,
(4.19)
whose boundary is given by [−L, L] × {0}, {−L} × [y(q, −T1 (q)), 0], {L} × [y(q, T2 (q)), 0], and by 1(q), the support of the curve s / → (x(q, s), y(q, s)). (See Figure 2.) Set
l=
0 1 L , y
where [·] denotes the integer part. For every k = 1, . . . , l, define % π $ qk = e 2 g e(k−1)yf (q0 ) ,
(4.20)
(4.21)
and, correspondingly, Qk = QL,d (qk ), σk (·) = σ (qk , ·), [−T1,k , T2,k ] = [−T1 (qk ), T2 (qk )], Wk = W (qk ) and 1k = 1(qk ). Let, moreover, 10 be the segment [−L, L] × {0} contained in W1 . For k ∈ {1, . . . , l − 1}, for every s ∈ (−L, L) ∩ (−T1,k , T2,k ), we want to identify the points (xk (s), yk (s)) ∈ 1k ⊂ Wk and (s, 0) ∈ Wk+1 . By construction, there exist a neighborhood U1k
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FIGURE 2
of (xk (s), yk (s)) in Qk , a neighborhood U2k of (s, 0) in Qk+1 , and an isometry ιk : U1k → U2k (with respect to the metric induced by M) such that ιk (U1k ∩ ∂Wk ) = U2k ∩ ∂Wk+1 . Consider the Riemannian two-dimensional manifold with boundary D(q0 ), obtained from the abstract union ∪1≤k≤l Wk by identification, for every k ∈ {1, . . . , l − 1} and every s ∈ [−L, L] ∩ [−T1,k , T2,k ], of (xk (s), yk (s)) ∈ Wk with (s, 0) ∈ Wk+1 . Since the gluing of two adjacent strips is rendered isometric (by the isometries ιk ), then the Riemannian structures induced by M on each of the strips Wk actually define a global Riemannian structure on D(q0 ). Let us assume, from now on, that
"
D(q0 )
K dA ≤ δ .
(4.22)
4.2. The fundamental tessellation Our purpose is to define a tessellation on D(q0 ), that is, a subdivision by geodesic polygons, in a checked pattern. Any such subdivision can be seen as a discrete system of coordinates globally defined on D(q0 ), assigning to any point the polygon which contains it.
A tessellation is determined by a grid of vertical and horizontal lines. The candidate vertical lines are the curves 1k defined above. Next lemma provides the estimates which allow to complete the desired construction.
Lemma 4.4. There exist δ0% ∈ (0, δ0 ), depending on K∞ , ε, and r, and two constants C(K∞ )
and C(ε, r) > 0 such that the following holds: Let q ∈ N verifies (4.16) with δ ≤ δ0% . Take t0 ∈ R, θ0 ∈ S 1 such that |t0 | ≤ L − y2 and |θ0 + π2 | ≤ δ. Define z0 = (t0 , 0, −θ0 ). Let (x(·), y(·), θ (·)) be the coordinates of σz0 (·) and denote by [0, T ] its maximal interval of definition in U W (q). Then (x(T ), & y(T )) ∈& 1(q) and |T − y| ≤ C(ε, r)δ. Moreover, for every s ∈ [0, T ], |x(s)| + |y(s) + s| + &θ (s) + π2 & ≤ C(K∞ )δ.
Proof of Lemma 4.4. every (x, y) ∈ W (q),
As it was done for estimates (3.5) and (3.6) of Section 3, we have, for 1 ≤ B(x, y) ≤ 1, 2
2 |F (x, y)| ≤ 2 K∞ .
Let I0 = [0, T0 ) be the largest interval (in [0, T ]) so that |θ (s) + π2 | < π3 for every s ∈ I0 . Since θ (0) = −θ0 , then T0 > 0. On I0 , the function v(s) = θ (s) + π/2 verifies the differential inequality 2 |v| ˙ ≤ 2 K∞ |v| , (4.23)
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which implies that & √ π && & &θ (s) + & ≤ δe2 K∞ s . 2
Since y˙ = − cos v and |x| ˙ ≤ 2| sin v|, we have, for every s ∈ I0 , " s √ v(τ )2 dτ ≤ δ 2 se4 K∞ s , |y(s) + s| ≤ 0
(4.24)
(4.25)
and |x(s)| ≤ 2δse2
√ K∞ s
.
(4.26)
Recall that, due to Lemma 4.3, the coordinates (x, y) of a point in 1(q) satisfy |y −y| ≤ 2Lδ. It follows from (4.24)–(4.26) that, if δ is small enough with respect to d, then T0 = T and σ (T ) ∈ 1(q). Moreover, |T − y| ≤ C(ε, r)δ. The estimates on x(·), y(·), and θ(·) follow. For every j ∈ {−l +1, . . . , l −1}, let 3j be the support of the geodesic in D(q0 ) which starts at (j y, 0) ∈ 10 ⊂ W1 , making an oriented angle −π/2 with 10 . Assume that (4.22) holds, with δ ≤ δ0% . For every j ∈ {−l + 1, . . . , l − 1}, a repeated application of Lemma 4.4 to 3j and the successive 1k shows that, for every k ∈ {0, . . . , l}, 3j intersects 1k transversally, provided that δ is small enough with respect to y. Indeed, at every step of the+iteration, the angle determined by 3j and 1k at their point of intersection differs from π/2 by Dj,k K dA ≤ δ, where Dj,k is the geodesic quadrilateral of D(q0 ) bounded by 30 , 3j , 10 , and 1k . For every j ∈ {−l +1, . . . , l −1} and k ∈ {0, . . . , l}, denote by zj,k the point of intersection of 3j and 1k . Due to Lemma 4.4, the length of the portion of 3j connecting zj,k with zj,k+1 differs from y by at most C(ε, r)δ. For the same reason, the length of the portion of 1k which joins zj,k and zj +1,k differs from y by at most C(ε, r)δ. Denote by Pj,k the geodesic quadrilateral with vertices zj,k , zj,k+1 , zj +1,k+1 , and zj +1,k , for (j, k) ∈ {−l + 1, . . . , l − 2} × {0, . . . , l − 1}. The edges of Pj,k are portions of the horizontal and vertical lines 3j , 3j +1 , 1k , and 1k+1 . The family of all such Pj,k is called a tessellation on D(q0 ).
Correspondingly, a tessellation on the Euclidean plane, which covers the rectangle [0, ly] × [−(l − 1)y, (l − 1)y], is given by the family of squares Cj,k = [j y, (j + 1)y] × [ky, (k + 1)y] ,
(j, k) ∈ {−l + 1, . . . , l − 2} × {0, . . . , l − 1} .
Define T as the union of all ∂Cj,k ⊂ R2 , that is, 3 4 3 4 l T = ∪l−1 j =−l+1 [0, ly] × {j y} ∪ ∪k=0 {ky} × [−(l − 1)y, (l − 1)y] ,
and, similarly, T % as the union of all ∂Pj,k ⊂ D(q0 ). It follows from the above considerations that there exists a homeomorphism 4 : T −→ T %
% , whose restriction to any edge of any square C such that 4(zj,k ) = zj,k j,k is (1 + C(K∞ , ε, r)δ)Lipschitz continuous.
Consider a teardrop of size εr , starting from the point (0, 0) ∈ R2 in the direction (1, 0), contained in the Euclidean rectangle [0, (l +1)y]×[−(l −1)y, (l −1)y]. Such teardrop trajectory
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FIGURE 3
meets T in a sequence of points pm = (xm , ym ), 0 ≤ m ≤ V , numbered according to the order of intersection. (See Figure 3.) Notice that p0 = pV = (0, 0) .
(4.27)
The idea is now to define the correspondents of the points pm on T % , to equip them with a direction, and to construct a control strategy steering q0 to q0− , passing through all these intermediate states.
4.3. Reduction to the elementary problem In the following discussion, it simplifies the presentation to assume that r/ε, the size of the teardrop, is large with respect to y, the tessellation step. If this is the case, then each portion of teardrop which is contained in a square of the tessellation, considered as a planar curve, presents a small variation in its angular component. To this extent, we ask ε to be small with respect to y, r being by hypothesis greater than one. Such requirement is not restrictive, because y, according to its definition (4.18), depends only on K∞ , and our final goal is to prove the complete controllability of (Dε ) for every positive ε. In detail, fix ω > 0 and assume that, for every r > 1, the teardrop of size r/ε intersects the grid of step y in a sequence of points pm , 0 ≤ m ≤ V (V depending on r), such that (a)
the total variation of the angle component of the portion of teardrop connecting pm with pm+1 is smaller than ω, for every 0 ≤ m < V .
Assume that ω > 0 is small enough, so that, from every sequence pm as above, we can extract a subsequence, still denoted by pm , 0 ≤ m ≤ V , verifying (b) (4.27) holds; (c) the Euclidean distance between pm and pm+1 is larger than y/2, for every 0 ≤ m < V ; (d) the portion of teardrop connecting pm with pm+1 intersects T at most at one more point, for every 0 ≤ m < V . Notice that (d) implies that
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
(a’)
583
the total variation of the angle component of the portion of teardrop connecting pm with pm+1 is smaller than 2ω, for every 0 ≤ m < V .
An sequence in T % is associated with {pm }Vm=0 by % pm = 4(pm ) .
Consider the broken geodesic in R2 obtained by connecting, for every 0 ≤ m < V , pm to pm+1 with a segment Sm . Such curve is characterized by the lengths |Sm | of the segments Sm and −1 by the family {αm }Vm=0 , where αm is the oriented angle between pm+1 − pm and pm+2 − pm+1 . According to (c) and (d), we have 0 1 y √ |Sm | ∈ , 5y , (4.28) 2 while an upper bound on αm follows from √ the remark that |αm | is maximal when Sm and Sm+1 are two concatenated cords of length 5y of a circle of radius r/ε. (See Figure 4.) By easy trigonometric considerations, 5√ 6 5 ε |αm | ≤ arcsin εy ≤ C(K∞ ) . (4.29) r r
FIGURE 4
A first approximation of the desired teardrop in D(q0 ) is obtained by constructing a broken % . geodesic connecting all points pm −1 % Lemma 4.5. There exists a family {Sm% }Vm=0 of geodesic segments in D(q0 ) such that Sm
% % and p % % % % connects pm m+1 , ||Sm | − |Sm || ≤ C(K∞ , ε, r)δ, and the angle αm between Sm and Sm+1 ε % satisfies |αm | ≤ C(K∞ ) r .
Proof of Lemma 4.5.
A principal coordinate strip in D(q0 ) is a portion of the domain D(q0 ) of the following type: Either it is bounded by two curves 3j and 3j +4 (and we denote it by
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6j ) or by two curves 1k and 1k+4 (we write 6 k ). In 6j (respectively, 6 k ) we can consider the geodesic coordinates with base line 3j +2 (respectively 1k+2 ). Let θm ∈ S 1 be defined by
eiθm =
pm+1 − pm . |pm+1 − pm |
% and p % Assume that −θ < θm ≤ θ or π − θ < θm ≤ π + θ and take j such that pm m+1 belong to π π 3π 3π 6j . (When 2 − θ < θm ≤ 2 + θ or 2 − θ < θm ≤ 2 + θ the same procedure can be carried % % % , y % ), (x % % out on a strip of the type 6 k .) Let (xm m m+1 , ym+1 ) be the coordinates of pm , pm+1 in 6j . It follows from Lemma 4.3, Lemma 4.4, and the Lipschitz continuity of 4 that &$ % & &$ % & % % % % & x & & & m+1 − xm − (xm+1 − xm ) + ym+1 − ym − (ym+1 − ym ) ≤ C(K∞ , ε, r)δ .
Therefore,
and so
& & % & y% & & m+1 − ym & − tan θ & % m & ≤ C(K∞ , ε, r)δ , % & xm+1 − xm &
& & 5 6 % % & & ym+1 − ym & & − θ &arctan m & ≤ C(K∞ , ε, r)δ . % % & & xm+1 − xm
% Lemma 4.3 can be used to estimate the coordinate behavior of geodesics starting from pm or, to be more precise, of solutions of the system (4.6)–(4.8) with initial conditions of the type % , y % , θ % ), with |θ % | far from π/2. By standard continuity considerations, there exists a geodesic (xm m % joining p % and p % % % % segment Sm m m+1 , whose initial direction is given by (xm , ym , θm ), with & & % &θ − θm & ≤ C(K∞ , ε, r)δ (4.30) m
and
&& % & & &&S & − |Sm |& ≤ C(K∞ , ε, r)δ . m
(4.31)
θm%
Moreover, we can assume that is the unique initial condition such that √ (4.31) holds. Indeed, geodesic segments satisfying (4.31) have no conjugate points, since π/ K ∞ is a uniform lower bound on the value of conjugate times. Therefore, Sm is defined independently of the choice of the principal coordinate strip. % between S % and S % In order to estimate the angle αm m m+1 , we notice that it is not restrictive % % % to assume that −θ < θm , θm+1 ≤ θ and that pm , pm+1 , and pm+2 belong to the same principal % % , such that coordinate strip 6j . Hence, there exists an angle θm+1 , which characterizes Sm+1 % |θm+1 − θm+1 | ≤ C(K∞ , ε, r)δ.
% is evaluated by θ % !% !% In the coordinate system of 6j , αm m+1 −θm , where θm is the angle coordinate % % θm% | ≤ δ. of the tangent vector to Sm at pm+1 . Notice that, according to Lemma 4.3, |θm% − ! The relation between the measure of an angle in coordinates and its Riemannian value has been discussed in Remark 2.1, where it was noticed that the conversion mapping is Lipschitz continuous, with Lipschitz constant depending continuously on the value of B. Since B is, by hypothesis, uniformly separated from zero, it follows that there exists a universal constant C > 0 such that & %& & & &α & ≤ C &θ % −! θ% & m
m+1
≤
≤
m
C|αm | + C(K∞ , ε, r)δ ε C(K∞ ) , r
Dubins’ Problem on Surfaces. I. Nonnegative Curvature
585
where the last equality follows from (4.29). Let βm be the oriented angle determined by the teardrop passing through pm and the segment Sm (with the agreement that the teardrop is oriented in its running sense and the segment from pm % ∈ U % D(q ) the unit vector which makes an angle β with S % . (When to pm+1 ). Denote by vm 0 m pm m % m = V , let vV be defined by (pV% , vV% ) = q0− .) In order to prove the existence of a trajectory in U D(q0 ) which connects q0 and q0− , admissible for the lifted Dubins problem, we will show % % % , v % ). The advantage is that all such , vm+1 ) is attainable from (pm that, for every m, (pm+1 m controllability subproblems, which we will call elementary problems, are essentially equivalent and that each of them “lives” in a single coordinate chart.
4.4. Solution of the elementary problem The m-th elementary problem is conveniently formulated in the geodesic coordinates of % . In such a coordinate system we have p % = (0, 0) and the rectangle whose base curve is Sm m % % % , the pm+1 = (|Sm |, 0). We noticed in Remark 2.1 that, at the points of the base curve Sm coordinate angle measures the true Riemannian angle between the corresponding unitary vector % . Therefore, what has to be solved is the motion planning problem defined by (2.8)–(2.10) and Sm % |, 0, α % + β with initial condition (0, 0, βm ) and final condition (|Sm m+1 ). m Fix a constant C0 = C(K∞ ) such that, & %& &α &, |βm | ≤ C0 ε m r
(4.32)
for every m.
In the Euclidean plane, the portion of teardrop connecting pm to pm+1 is a trajectory of x˙ = cos θ , y˙ = sin θ , u ∈ [−ε, ε] . (4.33) θ˙ = u , (x, y, θ )(0) = (0, 0, βm ),
. √ / Recall that |Sm | ∈ y2 , 5y . If ε is small enough with respect to d, then every solution of (4.33) intersects the surface , (x, y, θ ) ∈ R2 × S 1 | x = |Sm |
in a time close to |Sm |. Fix T > 0 such that every admissible trajectory of (4.33) intersects {x = |Sm |} only once, transversally, within time T . Let E(·) be the map which associates with an admissible control u : [0, T ] → [−ε, ε] the coordinates (y, θ ) of the trajectory corresponding to u(·), evaluated at the point of intersection with {x = |Sm |}. Notice that E is a $continuous % map from the space of admissible controls, endowed with the L1 topology, into R × − π2 , π2 .
The family of bang-bang control functions which are given by the concatenation of two arcs, the first one corresponding to control +ε and the second one to control −ε, form a continuous curve in L1 ([0, T ]), joining the two constant control functions u ≡ ε and u ≡ −ε. Taking into account also the two-bang concatenations where the controls are applied in the reversed order, it is clear that the family of all bang-bang control functions with at most two arcs forms a closed curve in the space of admissible controls. Choose a parameterization of such curve of$ the type % {us : [0, T ] → [−ε, ε]}s∈S 1 . Then γ : s / → E(us ) is a continuous closed curve in R × − π2 , π2 , which can be computed explicitly. In particular, straightforward computations show that there
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exists ρ = ρ(K∞ ) > 0 such that γ encircles the ball of center E(0) = (tan(βm )|Sm |, βm ) and radius ρε. Fix r = r(K∞ ) > 1 such that C0 ρ ≤ . r 6 Such a choice of r implies that γ encircles the segment & ' ε( & 6 = (0, θ) & |θ| ≤ 2C0 . r
Moreover, the distance from the support of γ to 6 is bounded from below by ρε/2, for δ small. Consider now the nonflat elementary problem θ x˙ = cos B , y˙ = sin θ, ˙ θ = u + F cos θ , (x, y, θ )(0) = (0, 0, β ) , m
u ∈ [−ε, ε] .
(4.34)
Fix any admissible control u : [0, T ] → [−ε, ε]. Denote by (x(·), y(·), θ (·)) (respectively, (x % (·), y % (·), θ % (·))) the solution of (4.33) [respectively, of (4.34)] corresponding to u(·). The same computations as in Lemma 4.3 imply that & & & & & & &x(t) − x % (t)& + &y(t) − y % (t)& + &θ (t) − θ % (t)& ≤ C(ε, K∞ )δ .
% In particular, we can assume that (x % (·), y % (·), θ % (·)), defined on [0, T ], intersects % |Sm |} once, $ π {xπ = % % % transversally. Define E (u) as the pair of coordinates (y (·), θ (·)) ∈ R × − 2 , 2 evaluated at the point of intersection. The map E % verifies & & &E(u) − E % (u)& ≤ C(ε, K∞ )δ .
Thus, the curve γ % : s / → E % (us ), which is closed and continuous, encircles 6, at least for δ small with respect to ε and 1/K∞ . By standard degree theory considerations, the image via E % of the space of admissible controls contains 6. Hence, the elementary problem is solvable, i.e., % % % , v % ) for the Dubins problem lifted on D(q ). It follows that (pm+1 , vm+1 ) is attainable from (pm 0 m q0− ∈ Aq0 and Theorem 4.1 is finally proved.
Acknowledgments The authors want to thank P. Pansu and K. Tapp for bringing to their attention interesting and useful results.
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Received August 12, 2005 Laboratoire des signaux et systèmes, Université de Paris-Sud, CNRS, Supélec, Gif-sur-Yvette, France INRIA, Sophia-Antipolis, France
Communicated by Robert Greene
