Optimal Control and Geodesics on Quadratic Matrix Lie Groups
Optimal Control and Geodesics on Quadratic Matrix Lie Groups Anthony M. Bloch† , Peter E. Crouch∗ , Jerrold E. Marsden♣ , and Amit K. Sanyal\ † Department of Mathematics University of Michigan Ann Arbor, MI 48109 [email protected] ∗ Department of Electrical Engineering \ Department of Mechanical Engineering University of Hawaii Honolulu, HI 96822 {pcrouch, aksanyal}@hawaii.edu ♣ Department of Control and Dynamical Systems California Institute of Technology Pasadena, CA 91125 [email protected] 17 January 2007
Abstract In this paper, we consider some matrix subgroups of the general linear group and in particular the special linear group that are defined by a quadratic matrix identity. The Lie algebras corresponding to these matrix groups include several classical semisimple matrix Lie algebras. We describe an optimal control problem on these groups which gives rise to geodesic flows with respect to a positive definite metric. These geodesic flows generalize the Euler equations and the symmetric representation of the geodesic flow for the n-dimensional rigid body (given earlier by Bloch, Crouch, Marsden and Ratiu), to these matrix groups. A discretization of these flows is given which gives a numerical algorithm for computation of the flow dynamics.
Contents 1 Introduction
2
2 Quadratic Matrix Groups
3
3 Optimal Control Problem 3.1 Extremal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Extremal Flow in Terms of an Involution . . . . . . . . . . . . . . . 3.3 A Conserved Quantity Along the Extremal Flow . . . . . . . . . . .
1
5 7 12 14
1 Introduction
2
4 Space of Extremal Solutions 4.1 Non-degeneracy Condition for the Canonical Symplectic Form . . . . 4.2 A Symplectic Submanifold of the Space of Extremal Solutions . . . . 4.3 Symplectic nature of the map from G × G to G × g . . . . . . . . . .
15 15 17 21
5 The Discrete Optimal Control Problem
22
6 Conclusions and Future Directions
23
1
Introduction
This paper treats an optimal control problem on some subgroups of the general and special linear group that satisfy a quadratic matrix identity. These groups include certain classical Lie groups and their corresponding Lie algebras. The extremal solutions to this optimal control problem are obtained as geodesic flows on the group with respect to a given positive definite metric, and are represented on the product space of the group with itself. These equations generalize the symmetric representation of the rigid-body equations given in Bloch, Crouch, Marsden, and Ratiu [2002] and some properties of rigid body dynamics. Variational and optimal control problems on Lie groups and symmetric spaces have been treated before in the literature. The Euler equations on the n-dimensional rigid body have been treated, for example, in Manakov [1976], Ratiu [1980], and Fedorov [1995], besides the symmetric representation in Bloch, Crouch, Marsden, and Ratiu [2002]. The symmetric representation of the generalized rigid body equations on the n-dimensional proper orthogonal group SO(n) (whose Lie algebra is denoted so(n)) is the set of equations Q˙ = QΩ; P˙ = P Ω. (1.1) The notation in these equations is as follows: The matrices Q and P are the dynamical variables, where Q ∈ SO(n) denotes the configuration of the body and P is a costate variable associated with the optimal control problem associated to these equations. For these equations to make sense as first order equations on SO(n) × SO(n), one needs to specify how Ω = Q−1 Q˙ ∈ so(n), the body angular velocity, is a function of Q and P , and thereby find a way to transform these equations to the usual Euler equations for the n-dimensional rigid body. This will will be explained in the body of the paper below. In this paper, we generalize these equations as well as the Euler equations for some matrix subgroups of the general and special linear group that are defined by a quadratic matrix identity. We introduce these matrix subgroups of and give some of their properties in Section 2. We introduce the optimal control problem on these groups in Section 3, and obtain the extremal solutions to this optimal control problem. In Section 3, we also obtain a map that transforms the symmetric representation to the Euler representation of the geodesic flows. In Section 4, we analyze the space of extremal solutions in the symmetric representation, and obtain a symplectic submanifold of this space that is invariant with respect to the extremal flows. In Section 5, we present some results on the discrete version of the optimal control problem, and the symmetric representation of the discrete extremal flows. Section 6 p resents some conclusions and discusses planned extensions of this work.
2 Quadratic Matrix Groups
2
3
Quadratic Matrix Groups
We consider quadratic matrix groups of the form n o G := g ∈ Rn×n | g T Jg = J ,
(2.1)
where g T is the transpose of the n × n matrix g, J 2 = αIn and J T = αJ for α = ±1. This class of groups includes standard classical groups of interest including the symplectic group and O(p, q) (see below), and enables one to generalize the symmetric representation of the rigid body flows discussed in Bloch, Crouch, Marsden, and Ratiu [2002]. Taking determinants in the relation g T Jg = J in (2.1), we see that for g ∈ G, det g = ±1. This class of matrix groups gives matrix representations of linear transformations on Rn that leave the following symmetric, bilinear form invariant: f (x, y) = xT Jy, x, y ∈ Rn . Thus, if g ∈ G, then f (gx, gy) = xT g T Jgy = xT Jy = f (x, y). Since J is not singular, the bilinear form f (x, y) is non-degenerate. The conditions following the definition (2.1) imply that J T = J −1 = αJ. Some examples of matrix groups covered by this definition are given below. Examples. 1. Choosing J = In (the n × n identity matrix) and α = 1, then G = O(n). 2. If n = 2m is even and
J=
0 Im , −Im 0
then the above definition gives the group Sp(2m) of 2m × 2m symplectic matrices. We remark that in the paper Bloch, Brinzanescu, Iserles, Marsden, and Ratiu [2006] we consider a related class of flows on symplectic group with J generalized to be an arbitrary skew-symmetric invertible matrix. Interestingly for integrability in that case however we do not consider the case of canoncical J. 3. If n = p + q and Ip 0 J= , 0 −Iq then the above definition gives the group O(p, q) of matrices that leave the non-degenerate, symmetric, bilinear form of signature (p, q) in Rn invariant. The Lie algebra of the group G is given by n o g = X ∈ Rn×n | X T J + JX = 0 . Note that for X ∈ g, J −1 X T J = −X and so Trace(X) = − Trace J −1 X T J = − Trace(X T ) = − Trace(X)
2 Quadratic Matrix Groups
4
and so Trace(X) = 0. Therefore, the Lie algebra g of G is a subalgebra of sl(n). The following statement constructs an element of the Lie algebra g from a given element in the group G. Lemma 2.1. If g ∈ G then g T ∈ G and g − g −1 ∈ g. Proof. Taking the matrix inverse of the relation g T Jg = J and using the fact that J −1 = αJ, we get J −1 = g −1 J −1 (g T )−1 ⇒ J = g −1 J(g T )−1 ⇒ (g T )−1 ∈ G ⇒ g T ∈ G. If g ∈ G, then g − g −1
T
J + J g − g −1 = g T J − Jg −1 + Jg − (g −1 )T J .
But from the relation g T Jg = J, we get g T J = Jg −1 and Jg = (g T )−1 J. Hence, we have g − g −1
T
J + J g − g −1 = 0,
and thus g − g −1 ∈ g.
The following statement combined with the previous statement leads to a trace orthogonal decomposition of the group G. Lemma 2.2. If g ∈ G and U ∈ g, then Trace(gU ) = − Trace(g −1 U ).
(2.2)
Proof. Using the identities J T = αJ = J −1 and U T J + JU = 0, we obtain Trace(gU ) = − Trace gJ −1 U T J = − Trace(JgJ −1 U T ) = − Trace U (J −1 )T g T J T = − Trace (J −1 )T g T J T U = − Trace α−1 J −1 g T αJU = − Trace J −1 g T JU = − Trace(g −1 U ), using the identity g T Jg = J in the last step.
3 Optimal Control Problem
5
Thus, for any g ∈ G, we have Trace (g + g −1 )U = 0, and hence, 1 G = (G + G−1 ) + 2 is a trace orthogonal decomposition of G. α = 1, G = SO(n), the above decomposition symmetric decomposition:
1 (G − G−1 ) 2 In the special case when J = In , is the standard symmetric plus skew-
1 1 Q = (Q + QT ) + (Q − QT ), 2 2 where Q ∈ SO(n), Q + QT is symmetric, and Q − QT ∈ so(n) is skew-symmetric. Note that if Λ = g + g −1 such that g ∈ G, then Λ satisfies JΛ − ΛT J = 0. This follows since if Λ = g + g −1 for some g ∈ G, then JΛ = J(g + g −1 ) = Jg + Jg −1 = (g T )−1 J + g T J T = g + g −1 J = ΛT J. In the next section, we will show how such a matrix Λ can be used to construct a metric for g. As a corollary of Lemma 2.2, we obtain the following result. Corollary 2.3. If g ∈ G and U, V ∈ g, then Trace(gV U ) = Trace(g −1 U V ).
(2.3)
Proof. We differentiate the identity (2.2) with respect to g in the direction gV . This gives us: Trace(gV U ) = Trace(g −1 gV g −1 U ) ⇔ Trace(gV U ) = Trace(g −1 U V ). These algebraic relations will be helpful in the analysis of the geodesic flows in the following section. In particular, they are useful in formulating a symmetric representation of the geodesic flows analogous to that for the generalized rigid body problem on SO(n).
3
Optimal Control Problem
In this section, we recall how to obtain geodesic flows using the maximum principle of optimal control theory (see Bloch, Ballieul, Crouch, and Marsden [2003];
3 Optimal Control Problem
6
Gelfand and Fomin [1963]; Kirk [2004]) and in particular, introduce an optimal control problem for the quadratic matrix group G. We use the trace pairing in gl(n): hA, Bi = Trace(AT B).
(3.1)
Let Σ : g → g be a fixed symmetric positive definite operator with respect to the inner product given by (3.1). Consider the optimal control problem on G given by T
Z min 0
1 hU, Σ(U )idt 4
(3.2)
subject to Q˙ = QU where U ∈ g, and where the minimum is taken over all curves Q(t) ∈ G with t ∈ [0, T ] and with fixed endpoints Q(0) = Q0 and Q(T ) = QT . Note that Σ defines a metric on g given by hU, V iΣ = hU, Σ(V )i = hΣ(U ), V i. Therefore, the problem (3.2) can be regarded as an optimal control problem with U ∈ g regarded as the control. This optimal control problem leads to an interesting form of the corresponding geodesic equations as described below. We have not yet specified the form of the symmetric positive definite operator Σ : g → g. This is done in the following result. Proposition 3.1. Let Λ, be a matrix satifying ΛT J = JΛ and such that Λ + ΛT is positive definite. Then Σ : g → g where Σ(U ) =
i 1h U (Λ + ΛT ) + (Λ + ΛT )U , 2
(3.3)
is a symmetric positive definite operator. Proof. We first show that Λ also satisfies ΛJ = JΛT . This is shown as follows: ΛJ = J T ΛT J J = J T ΛT J 2 = αJ T ΛT = JΛT . If Σ is as defined in equation (3.3), then U Λ + ΛU
T
J + J U Λ + ΛU
=ΛT U T J + U T ΛT J + JU Λ + JΛU =ΛT U T J + ΛT JU + U T JΛ + JU Λ =ΛT U T J + JU + U T J + JU Λ = 0, and U ΛT + ΛT U
T
J + J U ΛT + ΛT U
=ΛU T J + U T ΛJ + JU ΛT + JΛT U =ΛU T J + ΛJU + U T JΛT + JU ΛT =Λ U T J + JU + U T J + JU ΛT = 0.
3.1 Extremal flow
7
This shows that indeed Σ : g → g. For symmetricity and positive definiteness, we compute 1 hU, Σ(V )i = Trace U T V (Λ + ΛT ) + (Λ + ΛT )V 2 1 Trace V (Λ + ΛT )U T − U T (Λ + ΛT )V = 2 1 = Trace V ΛU T + U T ΛV + V ΛT U T + U T ΛT V 2 1 = Trace V ΛU T + JU ΛT V T J T + V ΛT U T + JU ΛV T J T 2 1 = Trace 2V ΛU T + 2V ΛT U T 2 = Trace V Λ + ΛT U T , which is symmetric and positive definite since Λ + ΛT is positive definite.
Thus, if Λ is J-symmetric (i.e., ΛT J = JΛ) and positive definite, then Σ : g → g as defined by (3.3) is a symmetric positive definite operator. Example: For the case G = Sp(2m), we take 0 Im Λ0 0 T . , Λ0 = Λ0 , and we have J = Λ= −Im 0 0 Λ0 This makes Λ J-symmetric, as 0 Λ0 Λ J= = JΛ. −Λ0 0 T
The condition Λ + ΛT is positive definite given in Proposition 3.1 is then equivalent to Λ0 is positive definite. This gives a choice for Σ when G = Sp(2m). Interestingly, this choice of Λ works even in the case that G = O(m, m), as can be easily verified using Proposition 3.1.
3.1
Extremal flow
Following the usual procedure of the Pontryagin maximum principle (see, for example, Bloch, Ballieul, Crouch, and Marsden [2003]), let P ∈ Rn×n denote the costate variable (consisting of Lagrange multipliers) used to impose the kinematic constraint Q˙ = QU . The Hamiltonian for the optimal control problem (3.2) is then defined as 1 H(P, Q, U ) = hP, QU i − hU, Σ(U )i 4 1 T = hQ P, U i − hU, Σ(U )i. 4
(3.4)
We obtain the extremal flow of this optimal control problem. The following result gives Hamilton’s equations for the Hamiltonian (3.4). Due to Pontryagin’s maximum principle, Hamilton’s equations for H give necessary conditions for a solution of the optimal control problem (3.2).
3.1 Extremal flow
8
Proposition 3.2. The necessary conditions for optimality of a solution to the optimal control problem (3.2) with costate P ∈ Rn×n yield the following Hamilton’s equations Q˙ = QU, P˙ = −P U T . (3.5)
Proof. The first of equations (3.5) is obtained from the kinematic constraint, corresponding to Q(t) ∈ G for t ∈ [0, T ]. The second of equations (3.5) is obtained as follows: P˙ = − gradQ H(P, Q, U ) = − gradQ hP, QU i = − gradQ hP U T , Qi = −P U T . Thus, we obtain the extremal flow given by (3.5).
To calculate the optimal control, we use Pontryagin’s maximum principle (see pg. 336 of Bloch, Ballieul, Crouch, and Marsden [2003]) and maximize H over U ∈ g. Thus, we solve the problem 1 1 (3.6) max hQT P, U i − hU, Σ(U )i + hΠ, U T J + JU i , U 4 2 where Π is a Lagrange multiplier for the constraint U T J + JU = 0. Now T U T J + JU = U T J T + J T U = α U T J + JU . So we choose Π to satisfy ΠT = αΠ. Then hΠ, U T J + JU i = hΠJ T , U T i + hJ T Π, U i = hJΠT , U i + hJ T Π, U i. But JΠT + J T Π = 2αJΠ, since ΠT = αΠ. So we have 1 hΠ, U T J + JU i = αhJΠ, U i. 2 Therefore, the necessary condition associated with (3.6) is 1 QT P − Σ(U ) + αJΠ = 0. 2
(3.7)
Note that U T J + JU = 0 ⇒ U ∈ g, and we want that Σ : g → g. Thus + αJΠ ∈ g, which means that
QT P
J(QT P + αJΠ) + (QT P + αJΠ)T J = 0 . Thus Π=−
1 JQT P + P T QJ . 2
Hence (3.7) becomes 1 α QT P − Σ(U ) − J JQT P + P T QJ = 0 2 2
(3.8)
3.1 Extremal flow
9
and Σ(U ) = QT P − αJP T QJ.
(3.9)
Now we need to check that QT P − αJP T QJ ∈ g. This is done below: J QT P − αJP T QJ + P T Q − αJ T QT P J T J = JQT P − P T QJ + P T QJ − JQT P = 0. Thus, we get U = Σ−1 QT P − αJP T QJ ∈ g. Hence the necessary conditions for the optimal control problem (3.2) is given by: Q˙ = QU, U = Σ−1 QT P − αJP T QJ , P˙ = −P U T ,
Q ∈ G, P ∈ Rn×n .
(3.10)
The above observation, that QT P − αJP T QJ ∈ g for any P ∈ Rn×n , is stated formally in the following result. Lemma 3.3. The map P : gl(n) → g given by P(A) =
1 A − αJAT J , 2
(3.11)
is a projection. Proof. We compute P2 (A) = P(P(A)) = = = =
T i 1h A − αJAT J − αJ A − αJAT J J 4 1 A − αJAT J − αJ(AT − αJ T AJ T )J 4 1 A − 2αJAT J + JJ T AJ T J 4 1 A − αJAT J = P(A), 2
since JJ T = J T J = αJ 2 = In . This proves the result.
Another interesting property of the map P : gl(n) → g is that it is self-dual. This is easily verified as follows; hB, P(A)i = Trace B T A − αJAT J T T = Trace AT B − α Trace JB JA
= Trace AT B − αJB T J = hA, P(B)i ⇒ P = P? .
Thus, if I denotes the identity map on gl(n), then I − P is the trace-orthogonal (to P) projection to the complement of g in gl(n). In terms of this projection, the optimal control in equation (3.10) can be expressed as U = 2Σ−1 P(QT P ) . (3.12)
3.1 Extremal flow
10
Note that we did not need to make any assumption on the costate P to prove Proposition 3.2. We may consider a reduced class of extremals derived from Proposition 3.2 by restricting the costate P to lie in G ⊂ Rn×n . The system (3.10) can be restricted naturally to the flow on G × G, since U, −U T ∈ g as given by (3.12). By doing so, we may lose some optimal trajectories of the variational/optimal control problem (3.2). Hence, QT P ∈ G, and therefore by Lemma 2.2, we have hQT P, U i = −h(QT P )−1 , U i. Thus, the Hamiltonian in this case can be written as 1 1 H(P, Q, U ) = hQT P − (QT P )−1 , U i − hU, Σ(U )i. 2 4
(3.13)
The following result is then obtained from Pontryagin’s maximum principle. Lemma 3.4. The extremal controls for the optimal control problem (3.2) when P ∈ G are given by Uext = Σ−1 QT P − (QT P )−1 . (3.14) −1 Proof. Since g = QT P ∈ G, P T QJ = J QT P . The expression for U in (3.10) now reads U = Σ−1 QT P − (QT P )−1 . Alternatively, a direct application of the maximum principle to the Hamiltonian in G × G (3.13) gives us gradU H(P, Q, U ) =
1 1 T Q P − (QT P )−1 − Σ(U ) = 0, 2 2
from whence we get the result.
The space Rn×n × Rn×n is a symplectic manifold with the canonical symplectic form Ωcan ((X1 , Y1 ), (X2 , Y2 )) = hY2 , X1 i − hY1 , X2 i. (3.15) Thus, the restriction of the costate P ∈ G ⊂ Rn×n is consistent, as shown in the proof of Lemma 3.4. This is stated formally in the following result. Proposition 3.5. The extremal flow (3.5) generated by the optimal control problem (3.2) which evolves on the canonical symplectic manifold (Rn×n × Rn×n , Ωcan ) as a Hamiltonian flow, naturally restricts to a flow on G × G. Here we identify Rn×n ×Rn×n with the cotangent bundle of Rn×n using the inner product (3.1), which then has a canonical cotangent structure. Let M = QT P − (QT P )−1 , then M ∈ g if P ∈ G by Lemma 2.1, in which case 1 H(P, Q, Uext ) = hM, Σ−1 (M )i, 4
(3.16)
and the extremal control can be expressed as Uext = Σ−1 (M ) ∈ g.
(3.17)
3.1 Extremal flow
11
We denote by Ω the restriction of Ωcan on G × G. Let S ⊂ G × G be an open submanifold containing the identity in G × G, where Ω is nondegenerate. Such a submanifold clearly exists as Ω is nondegenerate at the identity in G×G, since h·, ·i is nondegenerate on g. One can restrict the Hamiltonian (3.16) to S ⊂ G × G to show that the corresponding Hamiltonian flow is the same as the extremal flows (3.5) in Proposition 3.2. This is stated formally in the following result. Lemma 3.6. The extremal flows of the Hamiltonian (3.16) on S ⊂ G × G are also given by equations (3.5). Proof. Let X H denote the Hamiltonian vector field of this Hamiltonian, and let Z be another vector field on G × G. Thus, H dH(Z) = Ω(X H , Z) = hXQ , ZP i − hXPH , ZQ i.
If we let Z = (ZQ , ZP ) = (QV, P V ), then dH(QV ) = −hXPH , QV i,
H dH(P V ) = hXQ , P V i.
Using (2.3), the first of these expressions can be expressed as dH(QV ) = = = = =
E 1D E 1D T T V Q P, U + P −1 (QT )−1 (QV )T (QT )−1 , U 2 2 E 1D E 1D QV, P U T + (QT )−1 V T , (P −1 )T U 2 2 E 1 1D T QV, P U + Trace Q−1 (P −1 )T U V 2 2 E 1 1D QV, P U T + Trace(P T QV U ) from (2.3) 2 2 E 1D E D E 1D T QV, P U + QV, P U T = QV, P U T . 2 2
Similarly, the second expression can also be simplified as follows dH(P V ) = = = = =
E 1D E 1D T Q P V, U + P −1 (P V )P −1 (QT )−1 , U 2 2 E 1 1D hQU, P V i + V P −1 (QT )−1 , U 2 2 1 1 hQU, P V i + Trace Q−1 (P −1 )T V T U 2 2 1 1 hQU, P V i + Trace(P T QU V T ) from (2.3) 2 2 1 1 hQU, P V i + hQU, P V i = hQU, P V i . 2 2
H = QU . Hence, on the symplectic subset S of G×G, we get XPH = −P U T and XQ
A symplectic subset S of G × G, where the symplectic form Ω is non-degenerate, is determined in Section 4.
3.2 Extremal Flow in Terms of an Involution
3.2
12
Extremal Flow in Terms of an Involution
We now give another formulation of this extremal flow for the optimal control problem, in which we eliminate explicit reference to the transpose operator. We introduce on GL(n) (the Lie group of invertible n × n real matrices), the involution σ : GL(n) → GL(n);
σ(X) = (X T )−1 .
(3.18)
Thus, σ is an automorphism which satisfies σ 2 (·) = In , i.e., σ is an involution on GL(n). Clearly, by Lemma 2.1, σ restricts to an automorphism of G as well. The Lie group isomorphism σ induces a Lie algebra automorphism of g and gl(n), given by σ b : g → g; σ b(A) = −AT . (3.19) Hence, σ b2 (·) = In is also an involution on gl(n), and it is easy to directly check that g is mapped to itself by the involution σ b. If A ∈ g and X = exp(A) ∈ G where exp : g → G is the exponential map, then1 σ(X) = exp(−AT ) = exp (b σ (A)) . Now if Q˙ = QU , Q ∈ G and U ∈ g, then d σ(Q) =: σ∗ Q˙ = σ(Q)b σ (U ), dt
(3.20)
since Q˙ T = U T QT and (QT˙)−1 = −(QT )−1 U T . We can also express the extremal flows in terms of the relation (3.20) and d σ(P ) = σ? P˙ = (P T )−1 P˙ T (P T )−1 dt = (P T )−1 U = σ(P )U,
(3.21)
where we used equation (3.5) in the last step. Thus we can write M = QT P − (QT P )−1 = QT P − P −1 (QT )−1 which gives M = σ(Q−1 )P − P −1 σ(Q). Therefore, we can also express the extremal flow (3.5) as Q˙ = QU, P˙ = P σ b(U ), U = Σ−1 (M ), M = σ(Q−1 )P − P −1 σ(Q).
(3.22)
Equation (3.22) is an alternate way to express the extremal flow in terms of the involution σ of the Lie algebra g. We now use equations (3.22) to obtain the flow of M . Lemma 3.7. The flow of the quantity M along the extremal flow (3.22) is given by M˙ = [M, σ b(U )]. 1
(3.23)
According to Wolf [1972], σ b is a Cartan involution, since the fixed point set of σ b is so(n), which is the Lie algebra of a maximal compact subgroup of SL(n), the adjoint group of [gl(n), gl(n)] = sl(n).
3.2 Extremal Flow in Terms of an Involution
13
Proof. We have to evaluate d σ(Q−1 )P − P −1 σ(Q) . M˙ = dt Now σ(Q−1 ) = (σ(Q))−1 , so using (3.20), we get d σ(Q−1 ) = −b σ (U )σ(Q−1 ). dt Hence, along the extremal flow (3.22) of the optimal control problem, we have M˙ = −b σ (U )σ(Q−1 )P + σ(Q−1 )P σ b(U ) + σ b(U )P −1 σ(Q) − P −1 σ(Q)b σ (U ) = [σ(Q−1 )P − P −1 σ(Q), σ b(U )] = [M, σ b(U )], which proves the lemma.
The following statement summarizes these results as a generalization of the ndimensional rigid body (the Euler-Arnold) equations. Theorem 3.8. The ”generalized Euler” equations for the optimal control problem (3.2) are given by Q˙ = QU, M˙ = [M, σ b(U )], U = Σ−1 (M ).
(3.24)
These equations correspond to geodesic equations on G with the metric given by Σ. In the special case when G = SO(n), equations (3.24) give the Euler-Arnold equations, since σ b(U ) = U for U ∈ so(n). For other groups, these equations are still the usual co-adjoint equations on the dual of the Lie algebra. However, we use the trace pairing in gl(n) with a transpose as given by (3.1), unlike the texts Marsden and Ratiu [1998] and Arnold [1988]. Hence, if we chose M T instead of M to represent vectors on the dual of the Lie algebra, then the equation for M would be as given in these texts for the general group case. Note that if M satisfies the lax equation with −U T then M T satisfies the regular equation. Mischenko and Fomenko [1982] give the general form of Euler equations on semisimple Lie groups assuming the Hamiltonian function on T ? G is left-invariant. It is interesting to note that in the case of the orthogonal groups SO(n) one gets the same result either way. To pass between the formulations given by (3.22) and (3.24), we consider the map Φ : G × G → G × g, (Q, P ) 7→ (Q, M ) (3.25) where M = σ(Q−1 )P − P −1 σ(Q). Let us denote T −1 M g = Q P = exp sinh , 2 where the map sinh : g → g is defined by sinh X =
1 (exp(X) − exp(−X)) . 2
3.3 A Conserved Quantity Along the Extremal Flow
The inverse of the map Φ, where defined, is obtained simply by setting −1 M P = σ(Q) exp sinh , 2
14
(3.26)
Note that sinh(·) does indeed restrict to a map from g to g since if X ∈ g, exp(X) ∈ G, and hence exp(X) − exp(−X) ∈ g, by Lemma 2.1. In the case that J = In , α = 1 and G = SO(n), all equations (3.22)-(3.26) reduce to the symmetric representation of the rigid body equations in SO(n), given in Bloch, Crouch, Marsden, and Ratiu [2002], where the map sinh : so(n) → so(n) was first used.
3.3
A Conserved Quantity Along the Extremal Flow
From equations (3.5) or (3.22), we see that the quantity γ = P QT = P σ(Q)−1 is conserved along the extremal flow, since γ˙ = P˙ QT + P Q˙ T = −P U T QT + P U T QT = 0. We define the quantity m , γ − γ −1 = P QT − (P QT )−1 ,
(3.27)
which is also conserved along the extremal flow of the optimal control problem (3.2). If we now let m , γ = P QT = exp sinh−1 2 then we can express the costate as m P = exp sinh−1 σ(Q) = γσ(Q), (3.28) 2 which is an alternate expression to (3.26). The important difference between the expressions (3.26) and (3.28) is that γ ∈ G in the latter expression is a constant along the extremal flows. Therefore, the extremal costate P (t) can be expressed as the (matrix) product of a constant element γ in the group G with the involution of the extremal state Q(t) ∈ G. The constant γ ∈ G can be obtained from the initial state and costate as γ = P (0)Q(0)T . Now we can write γ = (QT )−1 (QT P )QT = σ(Q)gσ(Q)−1 , which implies that γ = Intσ(Q) g,
(3.29)
where for s, g ∈ G, the inner automorphism Ints (g) = sgs−1 is an analytic isomorphism of G onto itself (see Helgason [1978]). If we denote the algebra elements a = sinh−1
m M and A = sinh−1 , 2 2
4 Space of Extremal Solutions
15
then equation (3.29) can also be expressed as exp a = Intσ(Q) exp A = exp Adσ(Q) A , since the exponential map relates the inner automorphism on the group G with the adjoint action of the group on its algebra g (see Helgason [1978], Varadarajan [1984]). This shows that a = Adσ(Q) A ⇒ m = Adσ(Q) M,
(3.30)
which relates the quantities m and M through the adjoint action of the group G on its algebra g via the involution σ.
4
Space of Extremal Solutions
As we showed in the previous section, the extremal solutions to the optimal control problem (3.2) can be restricted to the space G × G, as a subspace of Rn×n × Rn×n . We now obtain a subset S of G×G that is a symplectic submanifold of Rn×n ×Rn×n , i.e., a subset in which the restriction of the canonical symplectic form Ωcan given by (3.15) is non-degenerate. The main result is stated below, and the following subsections show how this result is obtained. Theorem 4.1. The set S ⊂ G × G ⊂ Rn×n × Rn×n given by S , (Q, P ) ∈ G × G | m = P σ(Q−1 ) − σ(Q)P −1 , kmk < 2 ,
(4.1)
is a symplectic submanifold of Rn×n × Rn×n . The remainder of this section derives and proves this result.
4.1
Non-degeneracy Condition for the Canonical Symplectic Form
The restriction of the canonical symplectic form Ωcan to G × G is given by Ω((QA1 , P B1 ), (QA2 , P B2 )) = Ωcan |G×G ((QA1 , P B1 ), (QA2 , P B2 )) = hP B2 , QA1 i − hP B1 , QA2 i = Trace B2T P T QA1 − B1T P T QA2 .
(4.2)
We now identify submanifolds S of G × G restricted to which, Ω is nondegenerate. For a given point (Q, P ) ∈ S, non-degeneracy of (4.2) implies that we need to satisfy the following condition: if Trace B2T P T QA1 − B1T P T QA2 = 0 for all (A2 , B2 ) ∈ g × g, then (A1 , B1 ) = (0, 0). This is equivalent to requiring that for A ∈ g, Trace(B T P T QA) = 0 for all B ∈ g implies that A = 0. Using the projection P in Lemma 3.3, we know that for A ∈ gl(n), A = P(A) + (I − P)(A)
4.1 Non-degeneracy Condition for the Canonical Symplectic Form
16
where
1 1 A − αJAT J , (I − P)(A) = A + αJAT J . 2 2 ? Moreover since P = P , we have P(A) =
h(I − P)(A), P(B)i = hJ(I − P)(A), JP(B)i = h(P − P2 )(A), Bi = 0. Thus, gl(n) = JP gl(n) ⊕ J(I − P) gl(n) .
(4.3)
is an orthogonal decomposition. Note that T 1 1 α A − αJAT J J = α AT − αJAJ J 2 2 1 1 T = J JA J − αA = − αJ A − αJAT J 2 2 = −αJP(A)
T JP(A) =
and similarly T 1 α A + αJAT J J 2 1 = J JAT J + αA = 2 = αJ(I − P)(A).
T J(I − P)(A) =
1 = α AT + αJAJ J 2 1 αJ A + αJAT J 2
Also if A ∈ g then AT J = −JA = α(JA)T , so (JA)T = −α(JA), which corresponds to the fact that P : gl(n) → g. Now Trace B T P T QA = Trace (JB)T JP T QA D E = JB, JP T QA . We set g T = P T Q ∈ G, so that Trace B T P T QA = 0 is equivalent to hJB, Jg T Ai = 0. Now JB ∈ JP gl(n) for all B ∈ g, since P gl(n) = g. If hJB, Jg T Ai = 0 for all B ∈ g, then Jg T A ∈ J(I − P) gl(n) , according to the orthogonal decomposition (4.3). Therefore, we have Jg T A
T
= αJg T A
⇒ AT gJ
= Jg T A −1 T T ⇒ A = gT J A gJ.
But since A ∈ g, AT J = −JA or AT = −JAJ T , and therefore from above we obtain −1 T −1 T A = − gT J JAJ T gJ = − g T AJ gJ But g ∈ G, so g T Jg = J or g T
−1
= J T gJ, so (g T )−1 A(g T )−1 = −A, or
σ(g)Aσ(g) = −A.
(4.4)
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
17
If the above equation is not satisfied in some subset S of G × G for any non-zero A, then the symplectic form Ω is nondegenerate on that subset. We therefore have the following characterization of Smax , which is the maximal symplectic submanifold of G × G with the symplectic form induced from Ωcan : Smax = (Q, P ) ∈ G × G : g = QT P, σ(g)Aσ(g) = −A for A ∈ g ⇒ A = 0 . In the following subsection, we give a sufficient condition under which equation (4.4) implies that A = 0.
4.2
A Symplectic Submanifold of the Space of Extremal Solutions
We now determine a subset S ⊂ G × G which is symplectic, i.e., which satisfies equation (4.4) only when A = 0, for the symplectic form Ω. Now we give some basic results related to the matrix functions exp and sinh. Let the matrices Ψ and V be the matrices of eigenvalues and eigenvectors, respectively, of C; i.e. CV = V Ψ. Then we have: 1 2 1 3 exp(C)V = I + C + C + C + . . . V 2 3! 1 2 1 = V + CV + C V + C 3 V + . . . 2 3! 1 1 = V + V Ψ + V Ψ2 + V Ψ3 + . . . 2 3! 1 2 1 3 = V I + Ψ + Ψ + Ψ + . . . = V D, 2 3! where D = exp(Ψ). Similarly, it can be shown that exp(−C)V = V exp(−Ψ). Combining these two results, we obtain: 1 exp(C) − exp(−C) V 2 1 = V exp(Ψ) − exp(−Ψ) = V sinh Ψ. 2
(sinh C)V =
If B = sinh C, then this means that BV = V Λ, where Λ = sinh Ψ. We use the operator norm k · k on gl(n) defined by kAk = sup{kAxk | kxk = 1}. √ We know that the function sinh : C → C has an inverse sinh−1 (u) = ln u+ u2 + 1 with a convergent power series expansion for |u| < 1 (see Jeffrey [2000]). This power series expansion can be used to show that the map sinh : g → g has an inverse on the set U = {A ∈ g | kAk < 1}, and we denote this inverse sinh−1 : U → g as in Bloch, Crouch, Marsden, and Ratiu [2002]. The following lemma gives a sufficient condition under which equation (4.4) implies that A = 0 where A ∈ g.
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
Lemma 4.2. For A, B ∈ g, if kBk < 1 and exp sinh−1 B A exp sinh−1 B = −A,
18
(4.5)
then A = 0. Proof. Note that we do not make any assumptions on the diagonalizability of B ∈ g. Let {v1 , v2 , . . . , vm }, m ≤ n, be the set of independent eigenvectors of B, and let λk ∈ C denote the eigenvalue of B corresponding to the eigenvector vk ∈ Cn . Then we have (sinh−1 B)vk = (sinh−1 λk )vk , k = 1, . . . , m. If we denote g = exp sinh−1 B , then from the above equation we get gvk = dk vk , k = 1, . . . , m, where dk = exp sinh−1 λk ∈ C is the eigenvalue of g corresponding to the eigenvector vk . Now right-multiplication of equation (4.5) with vk gives gAgvk = −Avk ; that is, gA(dk vk ) = −Avk , i.e., dk (gfk ) = −fk . In other words, g −1 fk = −dk fk ,
(4.6)
where fk = Avk . Hence, the vector fk = Avk and the scalar quantity −dk form an eigenvector −1 −1 eigenvalue pair for g = exp − sinh B . But since g ∈ G is a non-singular matrix, g −1 vl = d1l vl , i.e., the eigenvalues of g −1 are also given by the reciprocals of the dl . Equation (4.6) is trivially satisfied if A = 0, since in that case fk = Avk = 0. For A 6= 0, we have at least one non-zero eigenvector-eigenvalue pair. Since the number of independent eigenvectors and the corresponding eigenvalues of g −1 are unique, equation (4.6) is equivalent to the presence of at least one pair (k, l) ∈ {1, . . . , m} such that (1 + dk dl ) = 0. Now if λk = rk exp(iθk ) are the eigenvalues of B, then the corresponding eigenvalues of C = sinh−1 B are sinh−1 λk = αk + iβk , where rk cos θk = sinh αk cos βk , rk sin θk = cosh αk sin βk , and dk = (exp αk )(cos βk + i sin βk ) are the eigenvalues of g. Thus, we have rk2 = sinh2 αk cos2 βk + cosh2 αk sin2 βk = sinh2 αk cos2 βk + (1 + sinh2 αk ) sin2 βk = sinh2 αk + sin2 βk . We know that kBk is the square root of the maximum eigenvalue of B T B, which is also the largest singular value of B; we also know that the largest absolute eigenvalue is less than or equal to the largest singular value Golub and Van Loan [1996]. Hence, if kBk < 1, then the eigenvalues of B satisfy |λk | = rk < 1 for all k ∈ {1, . . . , n}. Thus, from the previous expression for rk2 , we have sin2 βk = rk2 − sinh2 αk < 1 − sinh2 αk < 1 ⇒ | sin βk | < 1, which ensures that the principal value of βk ∈ (− π2 , π2 ). It is easy to verify that if |βk | < π2 for all k ∈ {1, . . . , n}, then dk dl = exp(αk + αl ) exp(i(βk + βl ))
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
19
is never a negative real number, and thus 1+dk dl 6= 0 for any pair (k, l) ∈ {1, . . . , n}. This implies that if equation (4.6) (alternatively (4.5)) is satisfied, then we must have F = 0, and thus A = 0. This completes the proof. From equation (4.4) and Lemma 4.2, we conclude that in the subset S˘ ⊂ G × G given by
σ b(M ) −1 −1 ˘
< 1 ⇒ kM k < 2 , S = (Q, P ) ∈ G × G | M = σ(Q )P − P σ(Q), 2 the symplectic form Ω is non-degenerate. Unfortunately, the dynamics of M given by (3.24) does not preserve kM k in general, since it evolves by conjugation by elements of G, and G may not be SO(n). Hence, the set SM as defined above is not an invariant set for the extremal (geodesic) flow. This difficulty is overcome by expressing the non-degeneracy condition or equation (4.4), in terms of the conserved quantity m, rather than M . To do this, we will utilize some of the relations developed in Section 3.3. From equation (3.29), we have g = σ(Q)−1 γσ(Q) = Intσ(Q)−1 γ. The involution of g is then obtained as σ(g) = σ Intσ(Q)−1 γ = σ σ(Q)−1 γσ(Q) o−1 n = σ(QT )γ T σ(QT )−1 = σ(QT )(γ T )−1 σ(QT )−1 , which yields σ(g) = Q−1 σ(γ)Q = IntQ−1 σ(γ).
(4.7)
Now we rewrite the non-degeneracy equation (4.4) replacing g with γ (and hence M with m) and Q, as follows: IntQ−1 σ(γ)A IntQ−1 σ(γ) = −A, which gives Q−1 σ(γ)QAQ−1 σ(γ)Q = −A; that is, σ(γ) AdQ Aσ(γ) = − AdQ A.
(4.8)
Thus, the non-degeneracy condition for the symplectic form Ω on G×G is equivalent to equation (4.8) implying that A = 0. Note that it is much easier to characterize a subset of G × G where Ω is non-degenerate using equation (4.8), rather than equation (4.4). This is because σ(γ) in (4.8) is conserved (since γ is conserved) along the extremal flow, unlike σ(g) in equation (4.4). Note also that one can transform between (Q, P ) and (Q, m) using equations (3.27) and (3.28). Now we are ready to prove the main result of this section, Theorem 4.1. Proof. (of Theorem 4.1). We can express σ(γ) in (4.8) terms of m as follows: b(m) −1 m −1 σ σ(γ) = σ exp sinh = exp sinh . 2 2
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
20
Note that since m is conserved along the extremal flow given by (3.22), the set S defined by equation (4.1) is also invariant with respect to this flow, and σ b(m) and γ are also conserved along this flow. Now applying Lemma 4.2, we conclude that if
σ
b(m) < 1
2 which means that kmk < 2, then AdQ A = 0 in equation (4.8), and hence A = 0 since Q ∈ G. Thus, Ω in the subset S is non-degenerate, and the result follows. We now show that if G = Sp(2n), then the norm bound giving the symplectic submanifold S ⊂ G × G in Theorem 4.1 is indeed a tight bound. To show this, we investigate the condition gBg = −B where g ∈ Sp(2n) and B ∈ sp(2n), and find an example where this condition is satisfied for B 6= 0. Now if 0 In X Y J= ,B = ∈ sp(2n), −In 0 Z W then Z = Z T , Y = Y T , and W = −X T , since B T J + JB = 0. If A B ∈ Sp(2n), g= C D then AT C = C T A, B T D = DT B, and AT D − C T B = In . Take g = J, then A = D = 0, B = In , C = −In . Now the condition gBg = −B requires X Y −W Z , =− Z W Y −X
X Y so W = X and Z = −Y . Also since B ∈ sp(2n) we have that B = −Y X T T where X = −X , Y = Y . This indeed gives us a non-zero B ∈ sp(2n) such that gBg = −B for g = J ∈ Sp(2n). We would like to express g = expsinh−1 m for m ∈ sp(2n). Note that 2 π π exp(αJ) = I2n cos α + J sin α. Thus exp 2 J = J and exp − 2 J = −J. Hence we have π 1 J = J − (−J) = J, sinh 2 2 and therefore we write π2 J = sinh−1 J noting that sinh−1 is multi-valued outside U ⊂ sp(2n), where sinh−1 is uniquely defined. Note also that J ∈ Sp(2n) ∩ sp(2n), and so J = exp sinh−1 J . Therefore we may identify m = 2J ∈ sp(2n) and J = exp sinh−1 m The condition in Lemma 4.2 states that if gBg = −g 2 ∈ Sp(2n). −1 m where g = exp sinh 2 and km/2k < 1, then B = 0. In this case, m = 2J and clearly km/2k = 1, since the spectral radius of J is 1 as all eigenvalues are ±i. Thus our choice of g = J is not included in the sufficient condition of Theorem 4.1 as g ∈ / S. Therefore we conclude that the norm bound provided in Theorem 4.1 for the symplectic submanifold S ⊂ G × G is tight.
4.3 Symplectic nature of the map from G × G to G × g
4.3
21
Symplectic nature of the map from G × G to G × g
Recall the map Φ : G × G → G × g given by (3.25). We can identify G × g with the cotangent bundle T ? G by the map ı : G × g → T ? G; ı(Q, M ) = Q, (QT )−1 M = Q, m(QT )−1 . (4.9) b = ı ◦ Φ : G × G → T ? G is symplectic. The We wish to demonstrate that the map Φ b is image of the submanifold S defined by (4.1) under Φ n o b Sm , Φ(S) = Q, m(QT )−1 ∈ T ? G | kmk < 2 ⊂ T ? G, (4.10) For g ∈ G, U, V ∈ g, using Corollary 2.3 we obtain the symplectic form on G × G as 1 −1 1 Ω (QU1 , −P V1T ), (QU2 , −P V2T ) = hU2 V1 −U1 V2 , QT P i+ hV1 U2 −V2 U1 , QT P i 2 2 (4.11) since hg, U V i = hg −1 , V U i. b : S ⊂ G × G → Sm ⊂ T ? G given by (3.25) and (4.9) is Theorem 4.3. The map Φ b ? ω = 2Ω. a symplectomorphism with Φ Proof. Let Z1 = (QU1 , −P V1T ) and Z2 = (QU2 , −P V2T ) be two vectors in G × G. We need to show that b ? Z1 , Φ b ? Z2 ) = 2Ω(Z1 , Z2 ). ω(Φ −1 d Now let N = M˙ = dt QT P − QT P , and −1 −1 −1 T d QT P = V T QT P − QT P U . dt Thus, for k = 1, 2, −1 −1 T − QT P Uk . Nk = UkT QT P − QT P VkT − VkT QT P Therefore, we obtain b ? Z1 , Φ b ? Z2 ) = hU1 , U T QT P − QT P V T − V T QT P −1 − QT P −1 U T i ω(Φ 2 2 2 2 −1 −1 T −hU2 , U1T QT P − QT P V1T − V1T QT P − QT P U1 i −1 +hQT P − QT P , [U1 , U2 ]i −1 −1 = h[U2 , U1 ], QT P i + h[U1 , U2 ], QT P i + hQT P − QT P , −1 T T [U1 , U2 ]i + hU2 V1 − U1 V2 , Q P i + hV1 U2 − V2 U1 , Q P i = 2Ω(Z1 , Z2 ), from equation (4.11). This proves the result.
5 The Discrete Optimal Control Problem
5
22
The Discrete Optimal Control Problem
In this section we briefly treat the discrete version of the continuous optimal control problem (3.2). We first define the discrete optimal control problem on the quadratic group G and then provide its solution. Let the matrix Λ satisfing ΛT J = JΛ, be such that Λ + ΛT is positive definite. Let Q0 , QN ∈ G be given fixed endpoints. We define the optimal control problem min Uk
N X
h∆, Uk i,
k=1
1 ∆ = (Λ + ΛT ), 2
(5.1)
Q0 = Q0 , QN = QN .
(5.2)
subject to Qk+1 = Qk Uk , Q−1 k Qk+1
Therefore Uk = ∈ G, and ∆ is positive definite satisfying the condition T ∆ J = ∆J = J∆. We note that we may restrict ourselves here to Uk close to the identity if we take the iteration procedure to be such that steps are small giving rise to small increments in Qk . The following result gives the solution to the discrete optimal control problem (5.1) subject to the constraints (5.2). Theorem 5.1. A solution of the discrete optimal control problem (5.1) is given by a sequence of matrices (Qk , Pk ) satisfying the optimal evolution equations Qk+1 = Qk Uk ,
Pk+1 = Pk σ(Uk ),
(5.3)
where σ : GL(n) → GL(n) is the involution defined by (3.18), and Uk is defined by −1 . (5.4) Uk ∆ − ∆Uk−1 = PkT Qk − PkT Qk Proof. First we form the discrete Hamiltonian for this discrete optimal control problem H(Pk+1 , Qk , Uk ) = hPk+1 , Qk Uk i − h∆, Uk i = hQT k Pk+1 − ∆, Uk i.
(5.5)
This Hamiltonian functional is linear in the controls, unlike the Hamiltonian (3.4) for the continuous optimal control problem. From this discrete Hamiltonian, we get T Pk = ∇Qk H = Uk Pk+1
T
= Pk+1 UkT ,
(5.6)
from which we obtain the discrete evolution equations (5.3). We need to find the critical points of H(Pk+1 , Qk , Uk ) where UkT JUk = J since Uk ∈ G. This Hamiltonian is of the form hAk , Uk i where Ak = QT k Pk+1 − ∆. Applying Pontryagin’s principle, we maximize the functional 1 Ha (Pk+1 , Qk , Uk ) = hAk , Uk i + hΠ, UkT JUk − Ji, 2 with respect to Uk , where Π is a Lagrange multiplier for the constraint UkT JUk = J. T Since UkT JUk − J = α UkT JUk − J , we require that ΠT = αΠ. Thus, we set the differential with respect to Uk to be zero: δUk Ha = hAk , δUk i + hαJUk Π, δUk i = 0.
6 Conclusions and Future Directions
23
Thus the gradient ∇Uk Ha = 0 when Ak + αJUk Π = QT k Pk+1 − ∆ + αJUk Π = 0, from which we get Π = −Uk−1 JAk . Utilizing the property that ΠT = αΠ, we get:
T −1 T Jσ(U ) = U J Q P − ∆ QT P − ∆ k k k+1 k k+1 k T ⇒ Pk+1 Qk − ∆ Uk J = JUkT QT P − ∆ . k k+1
T Q → QT P Equivalently, making the substitutions Uk → UkT , Pk+1 k k k+1 , we get:
QT P − ∆ UkT J k k+1 T ⇒ QT P − ∆U k k k J
T = JUk Pk+1 Qk − ∆ = J PkT Qk − Uk ∆ using (5.6)
⇒ JUk ∆ − ∆UkT J = JPkT Qk − QT k Pk J −1 −1 T ⇒ J Uk ∆ − ∆Uk = J Pk Qk − PkT Qk since Uk ∈ G and ∆J = J∆. This gives us the relation Uk ∆ − ∆Uk−1 = PkT Qk − PkT Qk
−1
,
(5.7)
from which we obtain Uk . The above result generalizes Theorem 6.5 in Bloch, Crouch, Marsden, and Ratiu [2002], which gives the symmetric representation of the extremal flows to the discrete optimal control problem in SO(n), to the quadratic groups G. Equation (5.4) can be written as −1 , (5.8) ΣD (Uk ) = PkT Qk − PkT Qk where ΣD : G → g, the discrete version of Σ, is defined by ΣD (U ) = U ∆ − ∆U −1 . Note that the derivative of ΣD at the identity is Σ, and since Σ is invertible, ΣD is a diffeomorphism from a neighborhood of the identity in G to a neighborhood of 0 in g. Therefore, ΣD can be inverted to obtain Uk from (5.8) as long as Uk remains in a neighborhood of the idendity. This value of Uk can then be used in a computational routine using the evolution equations (5.3) to update Qk and Pk . More generally in the Moser Veselov setting of SO(n) it is possible to find a global unique solution of Uk from (5.8). For details see Moser and Veselov [1991], Bloch, Crouch, Marsden, and Ratiu [2002], Cardoso and Leite [2001] and also Deift, Li and Tomei [1992]. We will discuss the general situation in a future publication.
6
Conclusions and Future Directions
In this paper we generalize the Euler equations and the symmetric representation of the n-dimensional rigid body equations to some quadratic matrix subgroups G of the special and general linear groups SL(n). These matrix groups are defined through a quadratic matrix identity. In addition, we describe the relationship between the symplectic structures on G × G and
REFERENCES
24
the cotangent bundle T ? G and find symplectic submanifolds of G × G and T ? G that are invariant with respect to the extremal flows. Results on the discrete version of the symmetric representation of the extremal flows on G × G are also presented. These discrete results can be used as a numerical algorithm to time-propagate the extremal flows. Further developments on the numerical side, including representation of the discrete extremal flows on G×g? which would generalize the Moser-Veselov equations to these quadratic groups, are among future research plans in this area. We also intend to analyze the complete integrability, and explore Lax pair formulations in the setting of symmetric representations of these extremal flows. In a future paper, we intend to work on an extension of this work to the special linear groups, and obtain the extremal flows of the optimal control problem on SL(n). This would generalize both the Euler equations and the symmetric representation of the extremal flows in the quadratic groups G to SL(n). Discrete versions of the extremal flows on SL(n) and their use as numerical algorithms to propagate these flows will also be investigated. These flows on the special linear group are interesting as models of deformable bodies and finite-dimensional models of fluids. Acknowledgements: Research supported in part by the National Science Foundation.
References Arnold, V. I. [1988], Dynamical Systems III, Encyclopedia of Mathematics 3, Springer-Verlag. Bloch, A., J. Ballieul, P. Crouch, and J. E. Marsden [2003], Nonholonomic Mechanics and Control, volume 24, Interdisciplinary Applied Mathematics, SpringerVerlag. Bloch, A. M., P. Crouch, J. E. Marsden, and T. S. Ratiu [2002], The symmetric representation of the rigid body equations and their discretization, Nonlinearity 15, 1309–1341. Bloch, A. M., Brinzanescu, V. A. Iserles, J. E. Marsden, and T. S. Ratiu [2006], A class of integrable geodesic flows on the symplectic group, preprint. Cardoso, J and Leite F. [2001] The Moser Veselov equation, Pre-Publicaos do Departamento de Matematic, Universidade de Coimbra. Deift, P. Li, L.-C. and Tomei, P. [1992] Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions, Mem. Am. Math. Soc. 100. Fedorov, Y. N. [1995], Various aspects of n-dimensional rigid body dynamics, American Mathematical Society Translations, 168, 141–171. Gelfand, I. M. and S. V. Fomin [1963], Calculus of Variations. (reprinted by Dover, 2000).
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Golub, G. H., and Van Loan, C. F. [1996], Matrix Computations, 3rd ed., the Johns Hopkins University Press, Baltimore, MD.
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Helgason, S. [1978], Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press. Humphreys, J. E. [1972], Introduction to Lie Algebra and Representation Theory, Springer- Verlag, New York, NY. Jeffrey, A. [2000], Inverse Trigonometric and Hyperbolic Functions §2.7 of Handbook of Mathematical Formulas and Integrals, 2nd ed., Academic Press, pp. 124–128, Orlando, FL. Kirk, D. E. [2004], Optimal Control Theory: An Introduction, Dover Publications, New York, NY. Manakov, S. V. [1976], Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body, Functional Analysis and its Applications, 10, 253–299. Marsden, J. E., and Ratiu, T. S. [1999], Introduction to Mechanics and Symmetry, Springer-Verlag, New York, NY, second edition. Mischenko, A. S., and Fomenko, A. T. [1982], Sel. Math. Sov., 2, 207–291. Moser, J. and Veselov, A. [1991], Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys. 139, 217-243. Ratiu, T. [1980], The Motion of the Free n-dimensional Rigid Body, Indiana University Mathematics Journal, 29, 609–629. Varadarajan, V. S. [1984], Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, New York, NY. Wolf, J. A. [1972], Spaces of Constant Curvature. Publish or Perish, Inc., Boston, MA, second edition.
Abstract In this paper, we consider some matrix subgroups of the general linear group and in particular the special linear group that are defined by a quadratic matrix identity. The Lie algebras corresponding to these matrix groups include several classical semisimple matrix Lie algebras. We describe an optimal control problem on these groups which gives rise to geodesic flows with respect to a positive definite metric. These geodesic flows generalize the Euler equations and the symmetric representation of the geodesic flow for the n-dimensional rigid body (given earlier by Bloch, Crouch, Marsden and Ratiu), to these matrix groups. A discretization of these flows is given which gives a numerical algorithm for computation of the flow dynamics.
Contents 1 Introduction
2
2 Quadratic Matrix Groups
3
3 Optimal Control Problem 3.1 Extremal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Extremal Flow in Terms of an Involution . . . . . . . . . . . . . . . 3.3 A Conserved Quantity Along the Extremal Flow . . . . . . . . . . .
1
5 7 12 14
1 Introduction
2
4 Space of Extremal Solutions 4.1 Non-degeneracy Condition for the Canonical Symplectic Form . . . . 4.2 A Symplectic Submanifold of the Space of Extremal Solutions . . . . 4.3 Symplectic nature of the map from G × G to G × g . . . . . . . . . .
15 15 17 21
5 The Discrete Optimal Control Problem
22
6 Conclusions and Future Directions
23
1
Introduction
This paper treats an optimal control problem on some subgroups of the general and special linear group that satisfy a quadratic matrix identity. These groups include certain classical Lie groups and their corresponding Lie algebras. The extremal solutions to this optimal control problem are obtained as geodesic flows on the group with respect to a given positive definite metric, and are represented on the product space of the group with itself. These equations generalize the symmetric representation of the rigid-body equations given in Bloch, Crouch, Marsden, and Ratiu [2002] and some properties of rigid body dynamics. Variational and optimal control problems on Lie groups and symmetric spaces have been treated before in the literature. The Euler equations on the n-dimensional rigid body have been treated, for example, in Manakov [1976], Ratiu [1980], and Fedorov [1995], besides the symmetric representation in Bloch, Crouch, Marsden, and Ratiu [2002]. The symmetric representation of the generalized rigid body equations on the n-dimensional proper orthogonal group SO(n) (whose Lie algebra is denoted so(n)) is the set of equations Q˙ = QΩ; P˙ = P Ω. (1.1) The notation in these equations is as follows: The matrices Q and P are the dynamical variables, where Q ∈ SO(n) denotes the configuration of the body and P is a costate variable associated with the optimal control problem associated to these equations. For these equations to make sense as first order equations on SO(n) × SO(n), one needs to specify how Ω = Q−1 Q˙ ∈ so(n), the body angular velocity, is a function of Q and P , and thereby find a way to transform these equations to the usual Euler equations for the n-dimensional rigid body. This will will be explained in the body of the paper below. In this paper, we generalize these equations as well as the Euler equations for some matrix subgroups of the general and special linear group that are defined by a quadratic matrix identity. We introduce these matrix subgroups of and give some of their properties in Section 2. We introduce the optimal control problem on these groups in Section 3, and obtain the extremal solutions to this optimal control problem. In Section 3, we also obtain a map that transforms the symmetric representation to the Euler representation of the geodesic flows. In Section 4, we analyze the space of extremal solutions in the symmetric representation, and obtain a symplectic submanifold of this space that is invariant with respect to the extremal flows. In Section 5, we present some results on the discrete version of the optimal control problem, and the symmetric representation of the discrete extremal flows. Section 6 p resents some conclusions and discusses planned extensions of this work.
2 Quadratic Matrix Groups
2
3
Quadratic Matrix Groups
We consider quadratic matrix groups of the form n o G := g ∈ Rn×n | g T Jg = J ,
(2.1)
where g T is the transpose of the n × n matrix g, J 2 = αIn and J T = αJ for α = ±1. This class of groups includes standard classical groups of interest including the symplectic group and O(p, q) (see below), and enables one to generalize the symmetric representation of the rigid body flows discussed in Bloch, Crouch, Marsden, and Ratiu [2002]. Taking determinants in the relation g T Jg = J in (2.1), we see that for g ∈ G, det g = ±1. This class of matrix groups gives matrix representations of linear transformations on Rn that leave the following symmetric, bilinear form invariant: f (x, y) = xT Jy, x, y ∈ Rn . Thus, if g ∈ G, then f (gx, gy) = xT g T Jgy = xT Jy = f (x, y). Since J is not singular, the bilinear form f (x, y) is non-degenerate. The conditions following the definition (2.1) imply that J T = J −1 = αJ. Some examples of matrix groups covered by this definition are given below. Examples. 1. Choosing J = In (the n × n identity matrix) and α = 1, then G = O(n). 2. If n = 2m is even and
J=
0 Im , −Im 0
then the above definition gives the group Sp(2m) of 2m × 2m symplectic matrices. We remark that in the paper Bloch, Brinzanescu, Iserles, Marsden, and Ratiu [2006] we consider a related class of flows on symplectic group with J generalized to be an arbitrary skew-symmetric invertible matrix. Interestingly for integrability in that case however we do not consider the case of canoncical J. 3. If n = p + q and Ip 0 J= , 0 −Iq then the above definition gives the group O(p, q) of matrices that leave the non-degenerate, symmetric, bilinear form of signature (p, q) in Rn invariant. The Lie algebra of the group G is given by n o g = X ∈ Rn×n | X T J + JX = 0 . Note that for X ∈ g, J −1 X T J = −X and so Trace(X) = − Trace J −1 X T J = − Trace(X T ) = − Trace(X)
2 Quadratic Matrix Groups
4
and so Trace(X) = 0. Therefore, the Lie algebra g of G is a subalgebra of sl(n). The following statement constructs an element of the Lie algebra g from a given element in the group G. Lemma 2.1. If g ∈ G then g T ∈ G and g − g −1 ∈ g. Proof. Taking the matrix inverse of the relation g T Jg = J and using the fact that J −1 = αJ, we get J −1 = g −1 J −1 (g T )−1 ⇒ J = g −1 J(g T )−1 ⇒ (g T )−1 ∈ G ⇒ g T ∈ G. If g ∈ G, then g − g −1
T
J + J g − g −1 = g T J − Jg −1 + Jg − (g −1 )T J .
But from the relation g T Jg = J, we get g T J = Jg −1 and Jg = (g T )−1 J. Hence, we have g − g −1
T
J + J g − g −1 = 0,
and thus g − g −1 ∈ g.
The following statement combined with the previous statement leads to a trace orthogonal decomposition of the group G. Lemma 2.2. If g ∈ G and U ∈ g, then Trace(gU ) = − Trace(g −1 U ).
(2.2)
Proof. Using the identities J T = αJ = J −1 and U T J + JU = 0, we obtain Trace(gU ) = − Trace gJ −1 U T J = − Trace(JgJ −1 U T ) = − Trace U (J −1 )T g T J T = − Trace (J −1 )T g T J T U = − Trace α−1 J −1 g T αJU = − Trace J −1 g T JU = − Trace(g −1 U ), using the identity g T Jg = J in the last step.
3 Optimal Control Problem
5
Thus, for any g ∈ G, we have Trace (g + g −1 )U = 0, and hence, 1 G = (G + G−1 ) + 2 is a trace orthogonal decomposition of G. α = 1, G = SO(n), the above decomposition symmetric decomposition:
1 (G − G−1 ) 2 In the special case when J = In , is the standard symmetric plus skew-
1 1 Q = (Q + QT ) + (Q − QT ), 2 2 where Q ∈ SO(n), Q + QT is symmetric, and Q − QT ∈ so(n) is skew-symmetric. Note that if Λ = g + g −1 such that g ∈ G, then Λ satisfies JΛ − ΛT J = 0. This follows since if Λ = g + g −1 for some g ∈ G, then JΛ = J(g + g −1 ) = Jg + Jg −1 = (g T )−1 J + g T J T = g + g −1 J = ΛT J. In the next section, we will show how such a matrix Λ can be used to construct a metric for g. As a corollary of Lemma 2.2, we obtain the following result. Corollary 2.3. If g ∈ G and U, V ∈ g, then Trace(gV U ) = Trace(g −1 U V ).
(2.3)
Proof. We differentiate the identity (2.2) with respect to g in the direction gV . This gives us: Trace(gV U ) = Trace(g −1 gV g −1 U ) ⇔ Trace(gV U ) = Trace(g −1 U V ). These algebraic relations will be helpful in the analysis of the geodesic flows in the following section. In particular, they are useful in formulating a symmetric representation of the geodesic flows analogous to that for the generalized rigid body problem on SO(n).
3
Optimal Control Problem
In this section, we recall how to obtain geodesic flows using the maximum principle of optimal control theory (see Bloch, Ballieul, Crouch, and Marsden [2003];
3 Optimal Control Problem
6
Gelfand and Fomin [1963]; Kirk [2004]) and in particular, introduce an optimal control problem for the quadratic matrix group G. We use the trace pairing in gl(n): hA, Bi = Trace(AT B).
(3.1)
Let Σ : g → g be a fixed symmetric positive definite operator with respect to the inner product given by (3.1). Consider the optimal control problem on G given by T
Z min 0
1 hU, Σ(U )idt 4
(3.2)
subject to Q˙ = QU where U ∈ g, and where the minimum is taken over all curves Q(t) ∈ G with t ∈ [0, T ] and with fixed endpoints Q(0) = Q0 and Q(T ) = QT . Note that Σ defines a metric on g given by hU, V iΣ = hU, Σ(V )i = hΣ(U ), V i. Therefore, the problem (3.2) can be regarded as an optimal control problem with U ∈ g regarded as the control. This optimal control problem leads to an interesting form of the corresponding geodesic equations as described below. We have not yet specified the form of the symmetric positive definite operator Σ : g → g. This is done in the following result. Proposition 3.1. Let Λ, be a matrix satifying ΛT J = JΛ and such that Λ + ΛT is positive definite. Then Σ : g → g where Σ(U ) =
i 1h U (Λ + ΛT ) + (Λ + ΛT )U , 2
(3.3)
is a symmetric positive definite operator. Proof. We first show that Λ also satisfies ΛJ = JΛT . This is shown as follows: ΛJ = J T ΛT J J = J T ΛT J 2 = αJ T ΛT = JΛT . If Σ is as defined in equation (3.3), then U Λ + ΛU
T
J + J U Λ + ΛU
=ΛT U T J + U T ΛT J + JU Λ + JΛU =ΛT U T J + ΛT JU + U T JΛ + JU Λ =ΛT U T J + JU + U T J + JU Λ = 0, and U ΛT + ΛT U
T
J + J U ΛT + ΛT U
=ΛU T J + U T ΛJ + JU ΛT + JΛT U =ΛU T J + ΛJU + U T JΛT + JU ΛT =Λ U T J + JU + U T J + JU ΛT = 0.
3.1 Extremal flow
7
This shows that indeed Σ : g → g. For symmetricity and positive definiteness, we compute 1 hU, Σ(V )i = Trace U T V (Λ + ΛT ) + (Λ + ΛT )V 2 1 Trace V (Λ + ΛT )U T − U T (Λ + ΛT )V = 2 1 = Trace V ΛU T + U T ΛV + V ΛT U T + U T ΛT V 2 1 = Trace V ΛU T + JU ΛT V T J T + V ΛT U T + JU ΛV T J T 2 1 = Trace 2V ΛU T + 2V ΛT U T 2 = Trace V Λ + ΛT U T , which is symmetric and positive definite since Λ + ΛT is positive definite.
Thus, if Λ is J-symmetric (i.e., ΛT J = JΛ) and positive definite, then Σ : g → g as defined by (3.3) is a symmetric positive definite operator. Example: For the case G = Sp(2m), we take 0 Im Λ0 0 T . , Λ0 = Λ0 , and we have J = Λ= −Im 0 0 Λ0 This makes Λ J-symmetric, as 0 Λ0 Λ J= = JΛ. −Λ0 0 T
The condition Λ + ΛT is positive definite given in Proposition 3.1 is then equivalent to Λ0 is positive definite. This gives a choice for Σ when G = Sp(2m). Interestingly, this choice of Λ works even in the case that G = O(m, m), as can be easily verified using Proposition 3.1.
3.1
Extremal flow
Following the usual procedure of the Pontryagin maximum principle (see, for example, Bloch, Ballieul, Crouch, and Marsden [2003]), let P ∈ Rn×n denote the costate variable (consisting of Lagrange multipliers) used to impose the kinematic constraint Q˙ = QU . The Hamiltonian for the optimal control problem (3.2) is then defined as 1 H(P, Q, U ) = hP, QU i − hU, Σ(U )i 4 1 T = hQ P, U i − hU, Σ(U )i. 4
(3.4)
We obtain the extremal flow of this optimal control problem. The following result gives Hamilton’s equations for the Hamiltonian (3.4). Due to Pontryagin’s maximum principle, Hamilton’s equations for H give necessary conditions for a solution of the optimal control problem (3.2).
3.1 Extremal flow
8
Proposition 3.2. The necessary conditions for optimality of a solution to the optimal control problem (3.2) with costate P ∈ Rn×n yield the following Hamilton’s equations Q˙ = QU, P˙ = −P U T . (3.5)
Proof. The first of equations (3.5) is obtained from the kinematic constraint, corresponding to Q(t) ∈ G for t ∈ [0, T ]. The second of equations (3.5) is obtained as follows: P˙ = − gradQ H(P, Q, U ) = − gradQ hP, QU i = − gradQ hP U T , Qi = −P U T . Thus, we obtain the extremal flow given by (3.5).
To calculate the optimal control, we use Pontryagin’s maximum principle (see pg. 336 of Bloch, Ballieul, Crouch, and Marsden [2003]) and maximize H over U ∈ g. Thus, we solve the problem 1 1 (3.6) max hQT P, U i − hU, Σ(U )i + hΠ, U T J + JU i , U 4 2 where Π is a Lagrange multiplier for the constraint U T J + JU = 0. Now T U T J + JU = U T J T + J T U = α U T J + JU . So we choose Π to satisfy ΠT = αΠ. Then hΠ, U T J + JU i = hΠJ T , U T i + hJ T Π, U i = hJΠT , U i + hJ T Π, U i. But JΠT + J T Π = 2αJΠ, since ΠT = αΠ. So we have 1 hΠ, U T J + JU i = αhJΠ, U i. 2 Therefore, the necessary condition associated with (3.6) is 1 QT P − Σ(U ) + αJΠ = 0. 2
(3.7)
Note that U T J + JU = 0 ⇒ U ∈ g, and we want that Σ : g → g. Thus + αJΠ ∈ g, which means that
QT P
J(QT P + αJΠ) + (QT P + αJΠ)T J = 0 . Thus Π=−
1 JQT P + P T QJ . 2
Hence (3.7) becomes 1 α QT P − Σ(U ) − J JQT P + P T QJ = 0 2 2
(3.8)
3.1 Extremal flow
9
and Σ(U ) = QT P − αJP T QJ.
(3.9)
Now we need to check that QT P − αJP T QJ ∈ g. This is done below: J QT P − αJP T QJ + P T Q − αJ T QT P J T J = JQT P − P T QJ + P T QJ − JQT P = 0. Thus, we get U = Σ−1 QT P − αJP T QJ ∈ g. Hence the necessary conditions for the optimal control problem (3.2) is given by: Q˙ = QU, U = Σ−1 QT P − αJP T QJ , P˙ = −P U T ,
Q ∈ G, P ∈ Rn×n .
(3.10)
The above observation, that QT P − αJP T QJ ∈ g for any P ∈ Rn×n , is stated formally in the following result. Lemma 3.3. The map P : gl(n) → g given by P(A) =
1 A − αJAT J , 2
(3.11)
is a projection. Proof. We compute P2 (A) = P(P(A)) = = = =
T i 1h A − αJAT J − αJ A − αJAT J J 4 1 A − αJAT J − αJ(AT − αJ T AJ T )J 4 1 A − 2αJAT J + JJ T AJ T J 4 1 A − αJAT J = P(A), 2
since JJ T = J T J = αJ 2 = In . This proves the result.
Another interesting property of the map P : gl(n) → g is that it is self-dual. This is easily verified as follows; hB, P(A)i = Trace B T A − αJAT J T T = Trace AT B − α Trace JB JA
= Trace AT B − αJB T J = hA, P(B)i ⇒ P = P? .
Thus, if I denotes the identity map on gl(n), then I − P is the trace-orthogonal (to P) projection to the complement of g in gl(n). In terms of this projection, the optimal control in equation (3.10) can be expressed as U = 2Σ−1 P(QT P ) . (3.12)
3.1 Extremal flow
10
Note that we did not need to make any assumption on the costate P to prove Proposition 3.2. We may consider a reduced class of extremals derived from Proposition 3.2 by restricting the costate P to lie in G ⊂ Rn×n . The system (3.10) can be restricted naturally to the flow on G × G, since U, −U T ∈ g as given by (3.12). By doing so, we may lose some optimal trajectories of the variational/optimal control problem (3.2). Hence, QT P ∈ G, and therefore by Lemma 2.2, we have hQT P, U i = −h(QT P )−1 , U i. Thus, the Hamiltonian in this case can be written as 1 1 H(P, Q, U ) = hQT P − (QT P )−1 , U i − hU, Σ(U )i. 2 4
(3.13)
The following result is then obtained from Pontryagin’s maximum principle. Lemma 3.4. The extremal controls for the optimal control problem (3.2) when P ∈ G are given by Uext = Σ−1 QT P − (QT P )−1 . (3.14) −1 Proof. Since g = QT P ∈ G, P T QJ = J QT P . The expression for U in (3.10) now reads U = Σ−1 QT P − (QT P )−1 . Alternatively, a direct application of the maximum principle to the Hamiltonian in G × G (3.13) gives us gradU H(P, Q, U ) =
1 1 T Q P − (QT P )−1 − Σ(U ) = 0, 2 2
from whence we get the result.
The space Rn×n × Rn×n is a symplectic manifold with the canonical symplectic form Ωcan ((X1 , Y1 ), (X2 , Y2 )) = hY2 , X1 i − hY1 , X2 i. (3.15) Thus, the restriction of the costate P ∈ G ⊂ Rn×n is consistent, as shown in the proof of Lemma 3.4. This is stated formally in the following result. Proposition 3.5. The extremal flow (3.5) generated by the optimal control problem (3.2) which evolves on the canonical symplectic manifold (Rn×n × Rn×n , Ωcan ) as a Hamiltonian flow, naturally restricts to a flow on G × G. Here we identify Rn×n ×Rn×n with the cotangent bundle of Rn×n using the inner product (3.1), which then has a canonical cotangent structure. Let M = QT P − (QT P )−1 , then M ∈ g if P ∈ G by Lemma 2.1, in which case 1 H(P, Q, Uext ) = hM, Σ−1 (M )i, 4
(3.16)
and the extremal control can be expressed as Uext = Σ−1 (M ) ∈ g.
(3.17)
3.1 Extremal flow
11
We denote by Ω the restriction of Ωcan on G × G. Let S ⊂ G × G be an open submanifold containing the identity in G × G, where Ω is nondegenerate. Such a submanifold clearly exists as Ω is nondegenerate at the identity in G×G, since h·, ·i is nondegenerate on g. One can restrict the Hamiltonian (3.16) to S ⊂ G × G to show that the corresponding Hamiltonian flow is the same as the extremal flows (3.5) in Proposition 3.2. This is stated formally in the following result. Lemma 3.6. The extremal flows of the Hamiltonian (3.16) on S ⊂ G × G are also given by equations (3.5). Proof. Let X H denote the Hamiltonian vector field of this Hamiltonian, and let Z be another vector field on G × G. Thus, H dH(Z) = Ω(X H , Z) = hXQ , ZP i − hXPH , ZQ i.
If we let Z = (ZQ , ZP ) = (QV, P V ), then dH(QV ) = −hXPH , QV i,
H dH(P V ) = hXQ , P V i.
Using (2.3), the first of these expressions can be expressed as dH(QV ) = = = = =
E 1D E 1D T T V Q P, U + P −1 (QT )−1 (QV )T (QT )−1 , U 2 2 E 1D E 1D QV, P U T + (QT )−1 V T , (P −1 )T U 2 2 E 1 1D T QV, P U + Trace Q−1 (P −1 )T U V 2 2 E 1 1D QV, P U T + Trace(P T QV U ) from (2.3) 2 2 E 1D E D E 1D T QV, P U + QV, P U T = QV, P U T . 2 2
Similarly, the second expression can also be simplified as follows dH(P V ) = = = = =
E 1D E 1D T Q P V, U + P −1 (P V )P −1 (QT )−1 , U 2 2 E 1 1D hQU, P V i + V P −1 (QT )−1 , U 2 2 1 1 hQU, P V i + Trace Q−1 (P −1 )T V T U 2 2 1 1 hQU, P V i + Trace(P T QU V T ) from (2.3) 2 2 1 1 hQU, P V i + hQU, P V i = hQU, P V i . 2 2
H = QU . Hence, on the symplectic subset S of G×G, we get XPH = −P U T and XQ
A symplectic subset S of G × G, where the symplectic form Ω is non-degenerate, is determined in Section 4.
3.2 Extremal Flow in Terms of an Involution
3.2
12
Extremal Flow in Terms of an Involution
We now give another formulation of this extremal flow for the optimal control problem, in which we eliminate explicit reference to the transpose operator. We introduce on GL(n) (the Lie group of invertible n × n real matrices), the involution σ : GL(n) → GL(n);
σ(X) = (X T )−1 .
(3.18)
Thus, σ is an automorphism which satisfies σ 2 (·) = In , i.e., σ is an involution on GL(n). Clearly, by Lemma 2.1, σ restricts to an automorphism of G as well. The Lie group isomorphism σ induces a Lie algebra automorphism of g and gl(n), given by σ b : g → g; σ b(A) = −AT . (3.19) Hence, σ b2 (·) = In is also an involution on gl(n), and it is easy to directly check that g is mapped to itself by the involution σ b. If A ∈ g and X = exp(A) ∈ G where exp : g → G is the exponential map, then1 σ(X) = exp(−AT ) = exp (b σ (A)) . Now if Q˙ = QU , Q ∈ G and U ∈ g, then d σ(Q) =: σ∗ Q˙ = σ(Q)b σ (U ), dt
(3.20)
since Q˙ T = U T QT and (QT˙)−1 = −(QT )−1 U T . We can also express the extremal flows in terms of the relation (3.20) and d σ(P ) = σ? P˙ = (P T )−1 P˙ T (P T )−1 dt = (P T )−1 U = σ(P )U,
(3.21)
where we used equation (3.5) in the last step. Thus we can write M = QT P − (QT P )−1 = QT P − P −1 (QT )−1 which gives M = σ(Q−1 )P − P −1 σ(Q). Therefore, we can also express the extremal flow (3.5) as Q˙ = QU, P˙ = P σ b(U ), U = Σ−1 (M ), M = σ(Q−1 )P − P −1 σ(Q).
(3.22)
Equation (3.22) is an alternate way to express the extremal flow in terms of the involution σ of the Lie algebra g. We now use equations (3.22) to obtain the flow of M . Lemma 3.7. The flow of the quantity M along the extremal flow (3.22) is given by M˙ = [M, σ b(U )]. 1
(3.23)
According to Wolf [1972], σ b is a Cartan involution, since the fixed point set of σ b is so(n), which is the Lie algebra of a maximal compact subgroup of SL(n), the adjoint group of [gl(n), gl(n)] = sl(n).
3.2 Extremal Flow in Terms of an Involution
13
Proof. We have to evaluate d σ(Q−1 )P − P −1 σ(Q) . M˙ = dt Now σ(Q−1 ) = (σ(Q))−1 , so using (3.20), we get d σ(Q−1 ) = −b σ (U )σ(Q−1 ). dt Hence, along the extremal flow (3.22) of the optimal control problem, we have M˙ = −b σ (U )σ(Q−1 )P + σ(Q−1 )P σ b(U ) + σ b(U )P −1 σ(Q) − P −1 σ(Q)b σ (U ) = [σ(Q−1 )P − P −1 σ(Q), σ b(U )] = [M, σ b(U )], which proves the lemma.
The following statement summarizes these results as a generalization of the ndimensional rigid body (the Euler-Arnold) equations. Theorem 3.8. The ”generalized Euler” equations for the optimal control problem (3.2) are given by Q˙ = QU, M˙ = [M, σ b(U )], U = Σ−1 (M ).
(3.24)
These equations correspond to geodesic equations on G with the metric given by Σ. In the special case when G = SO(n), equations (3.24) give the Euler-Arnold equations, since σ b(U ) = U for U ∈ so(n). For other groups, these equations are still the usual co-adjoint equations on the dual of the Lie algebra. However, we use the trace pairing in gl(n) with a transpose as given by (3.1), unlike the texts Marsden and Ratiu [1998] and Arnold [1988]. Hence, if we chose M T instead of M to represent vectors on the dual of the Lie algebra, then the equation for M would be as given in these texts for the general group case. Note that if M satisfies the lax equation with −U T then M T satisfies the regular equation. Mischenko and Fomenko [1982] give the general form of Euler equations on semisimple Lie groups assuming the Hamiltonian function on T ? G is left-invariant. It is interesting to note that in the case of the orthogonal groups SO(n) one gets the same result either way. To pass between the formulations given by (3.22) and (3.24), we consider the map Φ : G × G → G × g, (Q, P ) 7→ (Q, M ) (3.25) where M = σ(Q−1 )P − P −1 σ(Q). Let us denote T −1 M g = Q P = exp sinh , 2 where the map sinh : g → g is defined by sinh X =
1 (exp(X) − exp(−X)) . 2
3.3 A Conserved Quantity Along the Extremal Flow
The inverse of the map Φ, where defined, is obtained simply by setting −1 M P = σ(Q) exp sinh , 2
14
(3.26)
Note that sinh(·) does indeed restrict to a map from g to g since if X ∈ g, exp(X) ∈ G, and hence exp(X) − exp(−X) ∈ g, by Lemma 2.1. In the case that J = In , α = 1 and G = SO(n), all equations (3.22)-(3.26) reduce to the symmetric representation of the rigid body equations in SO(n), given in Bloch, Crouch, Marsden, and Ratiu [2002], where the map sinh : so(n) → so(n) was first used.
3.3
A Conserved Quantity Along the Extremal Flow
From equations (3.5) or (3.22), we see that the quantity γ = P QT = P σ(Q)−1 is conserved along the extremal flow, since γ˙ = P˙ QT + P Q˙ T = −P U T QT + P U T QT = 0. We define the quantity m , γ − γ −1 = P QT − (P QT )−1 ,
(3.27)
which is also conserved along the extremal flow of the optimal control problem (3.2). If we now let m , γ = P QT = exp sinh−1 2 then we can express the costate as m P = exp sinh−1 σ(Q) = γσ(Q), (3.28) 2 which is an alternate expression to (3.26). The important difference between the expressions (3.26) and (3.28) is that γ ∈ G in the latter expression is a constant along the extremal flows. Therefore, the extremal costate P (t) can be expressed as the (matrix) product of a constant element γ in the group G with the involution of the extremal state Q(t) ∈ G. The constant γ ∈ G can be obtained from the initial state and costate as γ = P (0)Q(0)T . Now we can write γ = (QT )−1 (QT P )QT = σ(Q)gσ(Q)−1 , which implies that γ = Intσ(Q) g,
(3.29)
where for s, g ∈ G, the inner automorphism Ints (g) = sgs−1 is an analytic isomorphism of G onto itself (see Helgason [1978]). If we denote the algebra elements a = sinh−1
m M and A = sinh−1 , 2 2
4 Space of Extremal Solutions
15
then equation (3.29) can also be expressed as exp a = Intσ(Q) exp A = exp Adσ(Q) A , since the exponential map relates the inner automorphism on the group G with the adjoint action of the group on its algebra g (see Helgason [1978], Varadarajan [1984]). This shows that a = Adσ(Q) A ⇒ m = Adσ(Q) M,
(3.30)
which relates the quantities m and M through the adjoint action of the group G on its algebra g via the involution σ.
4
Space of Extremal Solutions
As we showed in the previous section, the extremal solutions to the optimal control problem (3.2) can be restricted to the space G × G, as a subspace of Rn×n × Rn×n . We now obtain a subset S of G×G that is a symplectic submanifold of Rn×n ×Rn×n , i.e., a subset in which the restriction of the canonical symplectic form Ωcan given by (3.15) is non-degenerate. The main result is stated below, and the following subsections show how this result is obtained. Theorem 4.1. The set S ⊂ G × G ⊂ Rn×n × Rn×n given by S , (Q, P ) ∈ G × G | m = P σ(Q−1 ) − σ(Q)P −1 , kmk < 2 ,
(4.1)
is a symplectic submanifold of Rn×n × Rn×n . The remainder of this section derives and proves this result.
4.1
Non-degeneracy Condition for the Canonical Symplectic Form
The restriction of the canonical symplectic form Ωcan to G × G is given by Ω((QA1 , P B1 ), (QA2 , P B2 )) = Ωcan |G×G ((QA1 , P B1 ), (QA2 , P B2 )) = hP B2 , QA1 i − hP B1 , QA2 i = Trace B2T P T QA1 − B1T P T QA2 .
(4.2)
We now identify submanifolds S of G × G restricted to which, Ω is nondegenerate. For a given point (Q, P ) ∈ S, non-degeneracy of (4.2) implies that we need to satisfy the following condition: if Trace B2T P T QA1 − B1T P T QA2 = 0 for all (A2 , B2 ) ∈ g × g, then (A1 , B1 ) = (0, 0). This is equivalent to requiring that for A ∈ g, Trace(B T P T QA) = 0 for all B ∈ g implies that A = 0. Using the projection P in Lemma 3.3, we know that for A ∈ gl(n), A = P(A) + (I − P)(A)
4.1 Non-degeneracy Condition for the Canonical Symplectic Form
16
where
1 1 A − αJAT J , (I − P)(A) = A + αJAT J . 2 2 ? Moreover since P = P , we have P(A) =
h(I − P)(A), P(B)i = hJ(I − P)(A), JP(B)i = h(P − P2 )(A), Bi = 0. Thus, gl(n) = JP gl(n) ⊕ J(I − P) gl(n) .
(4.3)
is an orthogonal decomposition. Note that T 1 1 α A − αJAT J J = α AT − αJAJ J 2 2 1 1 T = J JA J − αA = − αJ A − αJAT J 2 2 = −αJP(A)
T JP(A) =
and similarly T 1 α A + αJAT J J 2 1 = J JAT J + αA = 2 = αJ(I − P)(A).
T J(I − P)(A) =
1 = α AT + αJAJ J 2 1 αJ A + αJAT J 2
Also if A ∈ g then AT J = −JA = α(JA)T , so (JA)T = −α(JA), which corresponds to the fact that P : gl(n) → g. Now Trace B T P T QA = Trace (JB)T JP T QA D E = JB, JP T QA . We set g T = P T Q ∈ G, so that Trace B T P T QA = 0 is equivalent to hJB, Jg T Ai = 0. Now JB ∈ JP gl(n) for all B ∈ g, since P gl(n) = g. If hJB, Jg T Ai = 0 for all B ∈ g, then Jg T A ∈ J(I − P) gl(n) , according to the orthogonal decomposition (4.3). Therefore, we have Jg T A
T
= αJg T A
⇒ AT gJ
= Jg T A −1 T T ⇒ A = gT J A gJ.
But since A ∈ g, AT J = −JA or AT = −JAJ T , and therefore from above we obtain −1 T −1 T A = − gT J JAJ T gJ = − g T AJ gJ But g ∈ G, so g T Jg = J or g T
−1
= J T gJ, so (g T )−1 A(g T )−1 = −A, or
σ(g)Aσ(g) = −A.
(4.4)
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
17
If the above equation is not satisfied in some subset S of G × G for any non-zero A, then the symplectic form Ω is nondegenerate on that subset. We therefore have the following characterization of Smax , which is the maximal symplectic submanifold of G × G with the symplectic form induced from Ωcan : Smax = (Q, P ) ∈ G × G : g = QT P, σ(g)Aσ(g) = −A for A ∈ g ⇒ A = 0 . In the following subsection, we give a sufficient condition under which equation (4.4) implies that A = 0.
4.2
A Symplectic Submanifold of the Space of Extremal Solutions
We now determine a subset S ⊂ G × G which is symplectic, i.e., which satisfies equation (4.4) only when A = 0, for the symplectic form Ω. Now we give some basic results related to the matrix functions exp and sinh. Let the matrices Ψ and V be the matrices of eigenvalues and eigenvectors, respectively, of C; i.e. CV = V Ψ. Then we have: 1 2 1 3 exp(C)V = I + C + C + C + . . . V 2 3! 1 2 1 = V + CV + C V + C 3 V + . . . 2 3! 1 1 = V + V Ψ + V Ψ2 + V Ψ3 + . . . 2 3! 1 2 1 3 = V I + Ψ + Ψ + Ψ + . . . = V D, 2 3! where D = exp(Ψ). Similarly, it can be shown that exp(−C)V = V exp(−Ψ). Combining these two results, we obtain: 1 exp(C) − exp(−C) V 2 1 = V exp(Ψ) − exp(−Ψ) = V sinh Ψ. 2
(sinh C)V =
If B = sinh C, then this means that BV = V Λ, where Λ = sinh Ψ. We use the operator norm k · k on gl(n) defined by kAk = sup{kAxk | kxk = 1}. √ We know that the function sinh : C → C has an inverse sinh−1 (u) = ln u+ u2 + 1 with a convergent power series expansion for |u| < 1 (see Jeffrey [2000]). This power series expansion can be used to show that the map sinh : g → g has an inverse on the set U = {A ∈ g | kAk < 1}, and we denote this inverse sinh−1 : U → g as in Bloch, Crouch, Marsden, and Ratiu [2002]. The following lemma gives a sufficient condition under which equation (4.4) implies that A = 0 where A ∈ g.
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
Lemma 4.2. For A, B ∈ g, if kBk < 1 and exp sinh−1 B A exp sinh−1 B = −A,
18
(4.5)
then A = 0. Proof. Note that we do not make any assumptions on the diagonalizability of B ∈ g. Let {v1 , v2 , . . . , vm }, m ≤ n, be the set of independent eigenvectors of B, and let λk ∈ C denote the eigenvalue of B corresponding to the eigenvector vk ∈ Cn . Then we have (sinh−1 B)vk = (sinh−1 λk )vk , k = 1, . . . , m. If we denote g = exp sinh−1 B , then from the above equation we get gvk = dk vk , k = 1, . . . , m, where dk = exp sinh−1 λk ∈ C is the eigenvalue of g corresponding to the eigenvector vk . Now right-multiplication of equation (4.5) with vk gives gAgvk = −Avk ; that is, gA(dk vk ) = −Avk , i.e., dk (gfk ) = −fk . In other words, g −1 fk = −dk fk ,
(4.6)
where fk = Avk . Hence, the vector fk = Avk and the scalar quantity −dk form an eigenvector −1 −1 eigenvalue pair for g = exp − sinh B . But since g ∈ G is a non-singular matrix, g −1 vl = d1l vl , i.e., the eigenvalues of g −1 are also given by the reciprocals of the dl . Equation (4.6) is trivially satisfied if A = 0, since in that case fk = Avk = 0. For A 6= 0, we have at least one non-zero eigenvector-eigenvalue pair. Since the number of independent eigenvectors and the corresponding eigenvalues of g −1 are unique, equation (4.6) is equivalent to the presence of at least one pair (k, l) ∈ {1, . . . , m} such that (1 + dk dl ) = 0. Now if λk = rk exp(iθk ) are the eigenvalues of B, then the corresponding eigenvalues of C = sinh−1 B are sinh−1 λk = αk + iβk , where rk cos θk = sinh αk cos βk , rk sin θk = cosh αk sin βk , and dk = (exp αk )(cos βk + i sin βk ) are the eigenvalues of g. Thus, we have rk2 = sinh2 αk cos2 βk + cosh2 αk sin2 βk = sinh2 αk cos2 βk + (1 + sinh2 αk ) sin2 βk = sinh2 αk + sin2 βk . We know that kBk is the square root of the maximum eigenvalue of B T B, which is also the largest singular value of B; we also know that the largest absolute eigenvalue is less than or equal to the largest singular value Golub and Van Loan [1996]. Hence, if kBk < 1, then the eigenvalues of B satisfy |λk | = rk < 1 for all k ∈ {1, . . . , n}. Thus, from the previous expression for rk2 , we have sin2 βk = rk2 − sinh2 αk < 1 − sinh2 αk < 1 ⇒ | sin βk | < 1, which ensures that the principal value of βk ∈ (− π2 , π2 ). It is easy to verify that if |βk | < π2 for all k ∈ {1, . . . , n}, then dk dl = exp(αk + αl ) exp(i(βk + βl ))
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
19
is never a negative real number, and thus 1+dk dl 6= 0 for any pair (k, l) ∈ {1, . . . , n}. This implies that if equation (4.6) (alternatively (4.5)) is satisfied, then we must have F = 0, and thus A = 0. This completes the proof. From equation (4.4) and Lemma 4.2, we conclude that in the subset S˘ ⊂ G × G given by
σ b(M ) −1 −1 ˘
< 1 ⇒ kM k < 2 , S = (Q, P ) ∈ G × G | M = σ(Q )P − P σ(Q), 2 the symplectic form Ω is non-degenerate. Unfortunately, the dynamics of M given by (3.24) does not preserve kM k in general, since it evolves by conjugation by elements of G, and G may not be SO(n). Hence, the set SM as defined above is not an invariant set for the extremal (geodesic) flow. This difficulty is overcome by expressing the non-degeneracy condition or equation (4.4), in terms of the conserved quantity m, rather than M . To do this, we will utilize some of the relations developed in Section 3.3. From equation (3.29), we have g = σ(Q)−1 γσ(Q) = Intσ(Q)−1 γ. The involution of g is then obtained as σ(g) = σ Intσ(Q)−1 γ = σ σ(Q)−1 γσ(Q) o−1 n = σ(QT )γ T σ(QT )−1 = σ(QT )(γ T )−1 σ(QT )−1 , which yields σ(g) = Q−1 σ(γ)Q = IntQ−1 σ(γ).
(4.7)
Now we rewrite the non-degeneracy equation (4.4) replacing g with γ (and hence M with m) and Q, as follows: IntQ−1 σ(γ)A IntQ−1 σ(γ) = −A, which gives Q−1 σ(γ)QAQ−1 σ(γ)Q = −A; that is, σ(γ) AdQ Aσ(γ) = − AdQ A.
(4.8)
Thus, the non-degeneracy condition for the symplectic form Ω on G×G is equivalent to equation (4.8) implying that A = 0. Note that it is much easier to characterize a subset of G × G where Ω is non-degenerate using equation (4.8), rather than equation (4.4). This is because σ(γ) in (4.8) is conserved (since γ is conserved) along the extremal flow, unlike σ(g) in equation (4.4). Note also that one can transform between (Q, P ) and (Q, m) using equations (3.27) and (3.28). Now we are ready to prove the main result of this section, Theorem 4.1. Proof. (of Theorem 4.1). We can express σ(γ) in (4.8) terms of m as follows: b(m) −1 m −1 σ σ(γ) = σ exp sinh = exp sinh . 2 2
4.2 A Symplectic Submanifold of the Space of Extremal Solutions
20
Note that since m is conserved along the extremal flow given by (3.22), the set S defined by equation (4.1) is also invariant with respect to this flow, and σ b(m) and γ are also conserved along this flow. Now applying Lemma 4.2, we conclude that if
σ
b(m) < 1
2 which means that kmk < 2, then AdQ A = 0 in equation (4.8), and hence A = 0 since Q ∈ G. Thus, Ω in the subset S is non-degenerate, and the result follows. We now show that if G = Sp(2n), then the norm bound giving the symplectic submanifold S ⊂ G × G in Theorem 4.1 is indeed a tight bound. To show this, we investigate the condition gBg = −B where g ∈ Sp(2n) and B ∈ sp(2n), and find an example where this condition is satisfied for B 6= 0. Now if 0 In X Y J= ,B = ∈ sp(2n), −In 0 Z W then Z = Z T , Y = Y T , and W = −X T , since B T J + JB = 0. If A B ∈ Sp(2n), g= C D then AT C = C T A, B T D = DT B, and AT D − C T B = In . Take g = J, then A = D = 0, B = In , C = −In . Now the condition gBg = −B requires X Y −W Z , =− Z W Y −X
X Y so W = X and Z = −Y . Also since B ∈ sp(2n) we have that B = −Y X T T where X = −X , Y = Y . This indeed gives us a non-zero B ∈ sp(2n) such that gBg = −B for g = J ∈ Sp(2n). We would like to express g = expsinh−1 m for m ∈ sp(2n). Note that 2 π π exp(αJ) = I2n cos α + J sin α. Thus exp 2 J = J and exp − 2 J = −J. Hence we have π 1 J = J − (−J) = J, sinh 2 2 and therefore we write π2 J = sinh−1 J noting that sinh−1 is multi-valued outside U ⊂ sp(2n), where sinh−1 is uniquely defined. Note also that J ∈ Sp(2n) ∩ sp(2n), and so J = exp sinh−1 J . Therefore we may identify m = 2J ∈ sp(2n) and J = exp sinh−1 m The condition in Lemma 4.2 states that if gBg = −g 2 ∈ Sp(2n). −1 m where g = exp sinh 2 and km/2k < 1, then B = 0. In this case, m = 2J and clearly km/2k = 1, since the spectral radius of J is 1 as all eigenvalues are ±i. Thus our choice of g = J is not included in the sufficient condition of Theorem 4.1 as g ∈ / S. Therefore we conclude that the norm bound provided in Theorem 4.1 for the symplectic submanifold S ⊂ G × G is tight.
4.3 Symplectic nature of the map from G × G to G × g
4.3
21
Symplectic nature of the map from G × G to G × g
Recall the map Φ : G × G → G × g given by (3.25). We can identify G × g with the cotangent bundle T ? G by the map ı : G × g → T ? G; ı(Q, M ) = Q, (QT )−1 M = Q, m(QT )−1 . (4.9) b = ı ◦ Φ : G × G → T ? G is symplectic. The We wish to demonstrate that the map Φ b is image of the submanifold S defined by (4.1) under Φ n o b Sm , Φ(S) = Q, m(QT )−1 ∈ T ? G | kmk < 2 ⊂ T ? G, (4.10) For g ∈ G, U, V ∈ g, using Corollary 2.3 we obtain the symplectic form on G × G as 1 −1 1 Ω (QU1 , −P V1T ), (QU2 , −P V2T ) = hU2 V1 −U1 V2 , QT P i+ hV1 U2 −V2 U1 , QT P i 2 2 (4.11) since hg, U V i = hg −1 , V U i. b : S ⊂ G × G → Sm ⊂ T ? G given by (3.25) and (4.9) is Theorem 4.3. The map Φ b ? ω = 2Ω. a symplectomorphism with Φ Proof. Let Z1 = (QU1 , −P V1T ) and Z2 = (QU2 , −P V2T ) be two vectors in G × G. We need to show that b ? Z1 , Φ b ? Z2 ) = 2Ω(Z1 , Z2 ). ω(Φ −1 d Now let N = M˙ = dt QT P − QT P , and −1 −1 −1 T d QT P = V T QT P − QT P U . dt Thus, for k = 1, 2, −1 −1 T − QT P Uk . Nk = UkT QT P − QT P VkT − VkT QT P Therefore, we obtain b ? Z1 , Φ b ? Z2 ) = hU1 , U T QT P − QT P V T − V T QT P −1 − QT P −1 U T i ω(Φ 2 2 2 2 −1 −1 T −hU2 , U1T QT P − QT P V1T − V1T QT P − QT P U1 i −1 +hQT P − QT P , [U1 , U2 ]i −1 −1 = h[U2 , U1 ], QT P i + h[U1 , U2 ], QT P i + hQT P − QT P , −1 T T [U1 , U2 ]i + hU2 V1 − U1 V2 , Q P i + hV1 U2 − V2 U1 , Q P i = 2Ω(Z1 , Z2 ), from equation (4.11). This proves the result.
5 The Discrete Optimal Control Problem
5
22
The Discrete Optimal Control Problem
In this section we briefly treat the discrete version of the continuous optimal control problem (3.2). We first define the discrete optimal control problem on the quadratic group G and then provide its solution. Let the matrix Λ satisfing ΛT J = JΛ, be such that Λ + ΛT is positive definite. Let Q0 , QN ∈ G be given fixed endpoints. We define the optimal control problem min Uk
N X
h∆, Uk i,
k=1
1 ∆ = (Λ + ΛT ), 2
(5.1)
Q0 = Q0 , QN = QN .
(5.2)
subject to Qk+1 = Qk Uk , Q−1 k Qk+1
Therefore Uk = ∈ G, and ∆ is positive definite satisfying the condition T ∆ J = ∆J = J∆. We note that we may restrict ourselves here to Uk close to the identity if we take the iteration procedure to be such that steps are small giving rise to small increments in Qk . The following result gives the solution to the discrete optimal control problem (5.1) subject to the constraints (5.2). Theorem 5.1. A solution of the discrete optimal control problem (5.1) is given by a sequence of matrices (Qk , Pk ) satisfying the optimal evolution equations Qk+1 = Qk Uk ,
Pk+1 = Pk σ(Uk ),
(5.3)
where σ : GL(n) → GL(n) is the involution defined by (3.18), and Uk is defined by −1 . (5.4) Uk ∆ − ∆Uk−1 = PkT Qk − PkT Qk Proof. First we form the discrete Hamiltonian for this discrete optimal control problem H(Pk+1 , Qk , Uk ) = hPk+1 , Qk Uk i − h∆, Uk i = hQT k Pk+1 − ∆, Uk i.
(5.5)
This Hamiltonian functional is linear in the controls, unlike the Hamiltonian (3.4) for the continuous optimal control problem. From this discrete Hamiltonian, we get T Pk = ∇Qk H = Uk Pk+1
T
= Pk+1 UkT ,
(5.6)
from which we obtain the discrete evolution equations (5.3). We need to find the critical points of H(Pk+1 , Qk , Uk ) where UkT JUk = J since Uk ∈ G. This Hamiltonian is of the form hAk , Uk i where Ak = QT k Pk+1 − ∆. Applying Pontryagin’s principle, we maximize the functional 1 Ha (Pk+1 , Qk , Uk ) = hAk , Uk i + hΠ, UkT JUk − Ji, 2 with respect to Uk , where Π is a Lagrange multiplier for the constraint UkT JUk = J. T Since UkT JUk − J = α UkT JUk − J , we require that ΠT = αΠ. Thus, we set the differential with respect to Uk to be zero: δUk Ha = hAk , δUk i + hαJUk Π, δUk i = 0.
6 Conclusions and Future Directions
23
Thus the gradient ∇Uk Ha = 0 when Ak + αJUk Π = QT k Pk+1 − ∆ + αJUk Π = 0, from which we get Π = −Uk−1 JAk . Utilizing the property that ΠT = αΠ, we get:
T −1 T Jσ(U ) = U J Q P − ∆ QT P − ∆ k k k+1 k k+1 k T ⇒ Pk+1 Qk − ∆ Uk J = JUkT QT P − ∆ . k k+1
T Q → QT P Equivalently, making the substitutions Uk → UkT , Pk+1 k k k+1 , we get:
QT P − ∆ UkT J k k+1 T ⇒ QT P − ∆U k k k J
T = JUk Pk+1 Qk − ∆ = J PkT Qk − Uk ∆ using (5.6)
⇒ JUk ∆ − ∆UkT J = JPkT Qk − QT k Pk J −1 −1 T ⇒ J Uk ∆ − ∆Uk = J Pk Qk − PkT Qk since Uk ∈ G and ∆J = J∆. This gives us the relation Uk ∆ − ∆Uk−1 = PkT Qk − PkT Qk
−1
,
(5.7)
from which we obtain Uk . The above result generalizes Theorem 6.5 in Bloch, Crouch, Marsden, and Ratiu [2002], which gives the symmetric representation of the extremal flows to the discrete optimal control problem in SO(n), to the quadratic groups G. Equation (5.4) can be written as −1 , (5.8) ΣD (Uk ) = PkT Qk − PkT Qk where ΣD : G → g, the discrete version of Σ, is defined by ΣD (U ) = U ∆ − ∆U −1 . Note that the derivative of ΣD at the identity is Σ, and since Σ is invertible, ΣD is a diffeomorphism from a neighborhood of the identity in G to a neighborhood of 0 in g. Therefore, ΣD can be inverted to obtain Uk from (5.8) as long as Uk remains in a neighborhood of the idendity. This value of Uk can then be used in a computational routine using the evolution equations (5.3) to update Qk and Pk . More generally in the Moser Veselov setting of SO(n) it is possible to find a global unique solution of Uk from (5.8). For details see Moser and Veselov [1991], Bloch, Crouch, Marsden, and Ratiu [2002], Cardoso and Leite [2001] and also Deift, Li and Tomei [1992]. We will discuss the general situation in a future publication.
6
Conclusions and Future Directions
In this paper we generalize the Euler equations and the symmetric representation of the n-dimensional rigid body equations to some quadratic matrix subgroups G of the special and general linear groups SL(n). These matrix groups are defined through a quadratic matrix identity. In addition, we describe the relationship between the symplectic structures on G × G and
REFERENCES
24
the cotangent bundle T ? G and find symplectic submanifolds of G × G and T ? G that are invariant with respect to the extremal flows. Results on the discrete version of the symmetric representation of the extremal flows on G × G are also presented. These discrete results can be used as a numerical algorithm to time-propagate the extremal flows. Further developments on the numerical side, including representation of the discrete extremal flows on G×g? which would generalize the Moser-Veselov equations to these quadratic groups, are among future research plans in this area. We also intend to analyze the complete integrability, and explore Lax pair formulations in the setting of symmetric representations of these extremal flows. In a future paper, we intend to work on an extension of this work to the special linear groups, and obtain the extremal flows of the optimal control problem on SL(n). This would generalize both the Euler equations and the symmetric representation of the extremal flows in the quadratic groups G to SL(n). Discrete versions of the extremal flows on SL(n) and their use as numerical algorithms to propagate these flows will also be investigated. These flows on the special linear group are interesting as models of deformable bodies and finite-dimensional models of fluids. Acknowledgements: Research supported in part by the National Science Foundation.
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