Hybrid Kernels and Capture Basins for Impulse Constrained Systems
Hybrid Kernels and Capture Basins for Impulse Constrained Systems Patrick Saint-Pierre Centre of Recherche Viabilit´e, Jeux, Contrˆ ole Universit´e Paris IX - Dauphine [email protected]
Abstract. We investigate, for constrained controlled systems with impulse, the subset of initial positions contained in a set K from which starts at least one run viable in K - the hybrid viability kernel - eventually until it reaches a given closed target in finite time - the hybrid capture basin. We define a constructive algorithm which approximates this set. The knowledge of this set is essential for control problem since it provides viable hybrid feed-backs and viable runs. We apply this method for approximating the Minimal Time-to-reach Function in the presence of both constraints and impulses. Two examples are presented, the first deals with a dynamical system revealing the complexity of the structure of hybrid kernels, the second deals with a Minimal Time problem with impulses.
1
Introduction
We consider a dynamical impulse system describing the evolution of a state variable x ∈ X = Rn which, in response to some events, may switch between a continuous evolution and an impulse evolution. Switches are triggered when the state reaches a closed set C. During some periods the state is governed by a continuous evolution until it reaches some state x ∈ C where a reset to a new position x+ ∈ Φ(x) occurs. We will distinguish between solution which, when reaching C, either may be reset or keep going the continuous evolution, and strict solution which, when reaching C, is necessarily reset. Research efforts are devoted to the study of such systems mainly in extending the mathematical tools of the Control Theory (see [22, Zabczyk], [5, Bensoussan & Lions] and in developing methods for analysis, verification or control design for non linear hybrid systems (see for instance [20, Sastry], [19, Shaft & Schumacher], [21, Vaandrager & Van Schuppen], [2, Aubin]). On the other hand the study of controlled system with state constraints has been widely developed this last decade using the main concepts of Set-Valued Analysis (see [3, Aubin & Cellina], [4, Aubin & Frankowska]) and Numerical SetValued Analysis (see [16, Saint-Pierre]) which can be regarded as a “Tool Box” for viability theory and numerical viability (viability, equilibria, stability, reachability, controllability, (see [1, Aubin], [14, Quincampoix], [17, Saint-Pierre]), optimal control (minimal time-to-reach problem, crisis time function, infinite C.J. Tomlin and M.R. Greenstreet (Eds.): HSCC 2002, LNCS 2289, pp. 378–392, 2002. c Springer-Verlag Berlin Heidelberg 2002
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horizon optimal control (see [7, Cardaliaguet, Quincampoix & Saint-Pierre], [11, Doyen & Saint-Pierre]), differential games (discriminating and leadership kernels, conditional and guaranteed strategies, minimal hitting time function, robustness, qualitative analysis (see [13, Leitmann], [18, Seube], [8, Cardaliaguet, Quincampoix & Saint-Pierre], [10, Dordan]) and their applications to mathematical economics, finance, demography or biology as well as to enginery and automatics when taking into account constraints is unescapable. The scope of this paper is to bridge these two fields in order to take into account explicit or hidden constraints that may occur when studying hybrid systems and to explore how numerical schemes can be implemented. Let us recall that the Viability Kernel Algorithm originally designed for computing viability kernels has been widen to approximate the smallest lower semicontinuous subsolution of the Hamilton-Jacobi-Belmann equation (see [12, Frankowska], [7, Cardaliaguet, Quincampoix & Saint-Pierre]). We prove that viability techniques can be extended with minor restrictions to hybrid systems first for computing the Hybrid Kernel which broadens the concept of Viability Kernel, second for computing the Minimal Time-to-reach function as example of Value Function.
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Hybrid Viability Kernels
2.1
Definitions: Run, Hybrid, Strict Hybrid Solutions
X that we assume to be Mar1. Let us consider a set-valued map F : X chaud1 . The continuous evolution is described by the differential inclusion x (t) ∈ F (x(t)), for almost all t ∈ R+
(1)
We denote by SFc (x0 ) the set of all absolutely continuous solutions of (1) starting from x0 at time t0 = 0. We denote by θC (x(·)) the first time t ≥ 0 when the solution x(·) reaches C. 2. Let us consider a set-valued map Φ defined on a closed set C = Dom(Φ) with compact values and closed graph. The impulse evolution is described by the recursive inclusion (2) xn+1 ∈ Φ(xn ) 3. We denote by SΦd (x0 ) the set of all discrete solutions {x0 , x1 , ..., xk } of (2) starting from x0 ∈ C. If finite, k is the first index such that xk ∈ / C. Definition 2.1 We call run of an impulse system (F, Φ) a sequence of elements − → x (·) := {(τi , xi , xi (·))}i∈I ∈ (R+ ×X ×C(0, ∞; X))N, where τi is the ith cadence, xi is the ith reinitialization, with x0 = x0 , and xi (·) ∈ SFc (xi ) is the ith motive which is an almost continuous solution to (1) starting from xi at time 0 until time τi . 1
Upper semicontinuous with non empty convex compact values and linear growth.
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We denote by S(F,Φ) (x0 ) the set of runs starting from x0 . The set of indexes I = {0, 1, ..., n} ⊂ N can be finite (n < +∞) or infinite (n = +∞), satisfying ∀i < n, ∃xi (·) ∈ SFc (xi ) such that xi (τi ) ∈ C, xi+1 ∈ Φ(xi (τi )) If τi = 0, xi+1 ∈ Φ(xi ) and then xi (·) is defined on an interval of length 0. n We set T = i=0 τi . Let us introduce a virtual lapse of time δ > 0, sequences (ti )i∈I and (ϑi )i∈I given by t0 = ϑ0 = 0, ti+1 = ti +τi and ϑi+1 = ϑi +τi +δ. We have ϑi+1 = ti + iδ. With any t ∈ [0, T ] we associate it satisfying tit ≤ t < tit +1 and with any ϑ ∈ R we associate iϑ satisfying ϑiϑ ≤ ϑ < ϑiϑ +1 . → Definition 2.2 With a run − x (·) we associate the expanded hybrid solution as e the map ϑ → x (ϑ) defined by iϑ τi e x (ϑ) = x0 + xi (τ )dτ + αi (xi+1 − xi (τi )) i=0
0
where the motive xi (τ ) is a solution to (1) starting from xi at time 0 until time τi , τi = min(τi , ϑ − ϑi ) and αi = max(0, min(1, ϑ − ϑi − τi )). We call hybrid → solution associated with a run − x (·) the map t → x(t) given by x(t) = xe (t + it δ). The expanded hybrid and the hybrid solutions are two parametrized representations of the same run. However, under slight assumption, there exists a one-to-one correspondence between expanded hybrid solutions and runs which fails when considering plain hybrid solutions. For studying impulse systems, that we call strict, when the solution is necessarily reset every times the trajectory reaches C to some position x∗ belonging to Φ(x∗ ), we need to introduce the following Definition 2.3 We call strict hybrid solution any hybrid solution such that ∀i ∈ I, xi (τi ) ∈ C and ∀τ ∈ [0, τi [, xi (τ ) ∈ / C. e e (F,Φ) (x0 ))) the set of extended (resp. strict) hybrid (x0 ) (resp. S We note S(F,Φ) solutions starting from x0 .
Let K be a closed set. We search the largest domain of initial positions from which starts at least one run remaining forever in K. We can assume without loss of generality that C = K ∩ Φ−1 (K) and we denote by S(F,Φ,K) (x0 ) the set → of runs − x (·) starting from x0 and viable in K. An impulse constrained system is characterized by the triple (F, Φ, K). The notion of hybrid kernel has been introduced in [2, Aubin]. It is characterized in terms of capture basin when K\C is a repeller2 . 2
We recall that a set D is a repeller if all solutions leave D in a finite time. The capture basin of a set C under constraint K for the dynamic F , denoted CaptF (C, K), is the set of initial position x0 from which there exists a solution x(·) ∈ SFc (x0 ) remaining in K until it reaches C in finite time.
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Definition 2.4 The hybrid (resp. strict hybrid) kernel of K for the impulse system (F, Φ, K) is the largest closed subset of initial states belonging to K from which starts at least one (resp. strict) hybrid viable solution. We denote this set Hyb(F,Φ) (K) (resp. Hyb (F,Φ) (K)). We have Hyb (F,Φ) (K) ⊂ Hyb(F,Φ) (K). We are going to prove that the hybrid kernel is closed and that hybrid and strict hybrid kernels can be approximated by a sequence of discrete viability kernels associated with suitable discrete systems. Assumptions and Approximated Impulse Systems We assume that • C is compact and Φ : C 0 < δinf ≤ inf
X is upper semi-continuous with compact values: inf d(x, y) ≤ sup sup d(x, y) ≤ δsup
x∈C y∈Φ(x)
(3)
x∈C y∈Φ(x)
so that Φ has no fix point and its graph is closed. We set ∀x ∈ / C, Φ(x) = ∅. • K is a compact set and F is a Marchaud map satisfying sup sup y ≤ M . x∈K y∈F (x)
This assumption implies that F is closed. As usual in the context of set-valued numerical analysis we need to consider “good” approximations Fρ of F in the sense that Graph(Fρ ) remains in a not too large neighborhood of F i) Graph(F ρ ) ⊂ Graph(F ) + φ(ρ)BX×X ii) F (x ) ⊂ Fρ (x)
(4)
x ∈B(x,ρM)
where BX×X denotes the unit ball of X × X and φ goes to zero when ρ. Let be h > 0 and set Ch := C + hB. With the set-valued map Φ we associate Φh (x) :=
Φ( x)
x e∈B(x,h)∩C
Proposition 2.5 If Φ is closed (its graph is closed) on C ×X, then Φh is closed. Proof —– Let (xn , y n ) ∈ Graph(Φh ) be a sequence converging to (x∗ , y ∗ ). From the definition of Φh , ∀n, ∃x en ∈ B(xn , h) ∩ C such that yn ∈ Φ(xen ). Since xen belongs to a compact subset and since C is closed, there exists a subsequence x en converging to ∗ ∗ ∗ x e ∈ B(x , h) ∩ C. Since Φ is closed on C, we have y ∈ Φ(xe) ⊂ Φh (x ) and consequently (x∗ , y ∗ ) ∈ Graph(Φh ).
Consistency of numerical schemes subordinates h to ρ in such a way that any element of x + ρFρ (x) sufficiently close to C belongs to Ch . We can choose3 h = ρ(M + φ(ρ)). 3
For computing viability kernels we refer to [16, Saint-Pierre].
(5)
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The Viability Kernel Algorithm
−−→ The discrete viability kernel of K for G denoted V iabG (K) is the largest closed subset of K of initial position from which there exists at least one sequence solution to the discrete dynamical system remaining in K: xn+1 ∈ G(xn ), ∀n ∈ N
(6)
The viability kernel algorithm consists in the construction of a decreasing sequence of subsets K n recursively defined by K 0 = K, and K n+1 = K n ∩ {x | G(x) ∩ K n = ∅} = K n ∩ G−1 (K n )
(7)
−−→ which converges to V iabG (K) (cf. [16, Saint-Pierre]): Proposition 2.6 Let G : X X be an upper semi-continuous and compact −−→ set valued map and K a compact set. Then Limn→∞ K n = V iabG (K). Let Fρ −−→ satisfying (4) and set Gρ = Id + ρFρ , then Limρ→0 V iabGρ (K) = V iabF (K) 2.3
Approximation Schemes for Impulse Systems
Let us fix a time step ρ and h = ρ(M + φ(ρ)). We replace the derivative x (t) of n+1 n x at time t by the difference x ρ−x where xn stands for x(nρ) with x0 = x(0).
n+1 We consider the set x = xn + ρϕn | ϕn ∈ Fρ (xn ) of successors of xn . We set if y ∈ / Ch y y if y ∈ / Ch Sρ (y) = and S ρ (y) = y ∪ Φh (y) if y ∈ ∂Ch y ∪ Φh (y) if y ∈ Ch Φh (y) if y ∈ Int(Ch ) Since Φh is closed, the graphs of set-valued maps Sρ and S ρ are also by construction closed. We consider the discrete dynamical systems i) xn+1 ∈ Gρ (xn ) = {Sρ (xn + ρϕ)| ϕ ∈ Fρ (xn )} ρ (xn ) = S ρ (xn + ρϕ)| ϕ ∈ Fρ (xn ) ii) xn+1 ∈ G
(8)
Proposition 2.7 If F and Φ are closed and if F is compact valued, then for all ρ are closed. ρ > 0, Gρ and G The proof is similar to the proof of Proposition 2.5. Let us choose ρ such that (9) 2M ρ < δinf We add to the discrete dynamical systems (8) i) and ii) two variables which evaluate a discrete time tn and the number of resets that took place until tn : n r if d(xn+1 , xn ) ≤ ρM tn+1 = tn + ρ and rn+1 = n r + 1 if d(xn+1 , xn ) ≥ δinf Thanks to assumption (3), evaluation of the distance between two successive states allows to distinguish between continuous and impulse evolutions and consequently to know whenever xn is or isn’t in a neighborhood of C.
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2.4
Impulse Viability Algorithm p and K p by 0 = K 0 = K. We define sequences K Let be K ρ ρ ρ ρ ρp+1 = {x ∈ K ρp | G ρ (x) ∩ K ρp = ∅}, Kρp+1 = {x ∈ Kρp | Gρ (x) ∩ Kρp = ∅} (10) K In other words, x ∈ Kρp+1 if and only if x ∈ Kρp and ∃ϕ ∈ Fρ (x) such that Sρ (x + ρϕ) ∩ Kρp = ∅ that is to say that x + ρϕ ∈ Sρ−1 (Kρp ) and (10) becomes: ρp+1 := (Id + ρFρ )−1 (S ρ−1 (K ρp )) ∩ K ρp , Kρp+1 := (Id + ρFρ )−1 (Sρ−1 (Kρp )) ∩ Kρp K ρp and Kρp Since the graphs of S ρ , Sρ and Id+ρFρ are closed and K is compact, K p p are compact. Decreasing sequences Kρ and Kρ converge in the sense of Painlev´e−→ −−→ ρ∗ = − Kuratowski to limit sets K ViabGeρ (K) and Kρ∗ = ViabGρ (K) satisfying ∗ = (1 + ρFρ )−1 (S −1 (K ∗ )) and K ∗ = (1 + ρFρ )−1 (S −1 (K ∗ )) K ρ ρ ρ ρ ρ ρ 2.5
(11)
Approximation of the Hybrid Viability Kernels
We have to prove that the limit Limρ→0 Kρ∗ exists and is equal to Hyb(F,Φ) (K) and that it remains true, under restrictive assumptions for strict impulse systems. This derives from the following properties which proofs are annexed. Proposition 2.8 Under assumptions (3) and (4), we have ∗ ⊂ Hyb(F,Φ) (K) Limsupρ→0 Kρ∗ ⊂ Hyb(F,Φ) (K) and Limsupρ→0 K ρ
(12)
and Proposition 2.9 Let us assume (3) and (4). Then Hyb(F,Φ) (K) ⊂ Kρ∗ . from which we deduce our main result Theorem 2.10 (Hybrid Kernel) Let us assume (3) and (4). Then Hyb(F,Φ) (K) = Limρ→0 Kρ∗ which is closed. Case of strict impulse system The difference between hybrid and strict hybrid solutions arises from that any strict solution cannot encounter C without being reset whereas hybrid solution may follow a continuous path even if it encounters C. The strict hybrid kernel is not necessarily closed as shown in the following example. Exemple Let us consider the impulse dynamical system given by F (x, y) = (−1, 0) Φ(x, y) = {x + 1} × {x + y + 1} (13) C = {(x, y) such that sup(|x|, |y|) = 1 and ((x ≤ 0) or (y ≥ 0))} K = [−2, 2] × [−2, 2] It is easy to check (see figure 1) that the strict hybrid kernel is not closed. We aim to give a sufficient condition implying the convergence of discrete ∗ to the strict hybrid kernel strict hybrid kernels K ρ r Let be ∂C := {x ∈ ∂C such that ∃px , T∂C (x) = {y | < px , y >= 0}}, the regular part of the boundary of C.
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Fig. 1. Closed hybrid and non closed strict hybrid kernels.
Proposition 2.11 (Strict Hybrid Kernel) Let us assume (3), (4), that F is Lipschitz on a neighborhood of C and that the two following properties hold true: (A0) ∀x ∈ ∂C r , F (x) ∩ T∂C (x) = ∅, (A1) ∀x ∈ ∂C\∂C r any solution of (1) starting from x leaves K in finite time without reaching C. ∗ Then Hyb (F,Φ) (K) = Limρ→0 Kρ and Hyb(F,Φ) (K) is closed.
e ρ∗ a sequence converging to some element x0 . From PropoProof —— Let be x0ρ ∈ K → x = {(τi , xi , xi (·)}i∈I sition 2.8, x0 ∈ Hyb(F,Φ) (K) and so there exists a viable run − →ρ = {(τρi , xρi , starting from x0 solution of system (F, Φ, K) and there exist viable runs − x xρi (·)}i∈I starting from x0ρ such that the sequence of associated expanded strict solutions xeρ (·) converge uniformly to xe (·) when ρ → 0. g (F,Φ)(K). Then, there exists j ∈ I such that xj (0) = xj , Assume that x0 ∈ / Hyb xj (τj ) ∈ C if τj < ∞ and ∃τbj ∈]0, τj [ such that xj (τbj ) ∈ ∂C. • If xj (τbj ) ∈ ∂C r , from assumption (A0) there exist η > 0 and γ > 0 such that < pxj (τbj ) , xj (τbj − η) − xj (τbj ) >< −γ and < pxj (τbj ) , xj (τbj + η) − xj (τbj ) >> γ. Using a continuity argument, since F is Lipschitz, ∃ρb such that ∀ρ ≤ ρb: ∃τbρj ∈ [τbj − η, τbj + η] ⊂]0, τρj [, xρj (τbρj ) ∈ ∂C →ρ is a x which is impossible since xρ (·) is the hybrid solution associated with the run − strict hybrid solution. • If xj (τbj ) ∈ ∂C\∂C r , then from assumption (A1) the hybrid solution x(·) associ→ ated with the viable run − x would leave K in finite time which is impossible.
We can replace assumption (A1) by a global assumption of emptyness of the viability kernel of K\C under F (see [2, Aubin]). Corollary 2.12 Let us assume (3) and (4). If K is a repeller for F and if no solution starting from C and immediately leaving C reaches C then ∗ Hyb (F,Φ) (K) = Limρ→0 Kρ .
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Proof ——- Since no solution starting and immediately leaving from C reaches C and, since K is a repeller for F , all solutions leave K in a finite time. So that any hybrid viable solution which encounters C at some position x∗ must necessarily be reset from x∗ to a position belonging to Φ(x∗ ) for maintaining viability.
2.6
Exemple of Numerical Computation of Hybrid Kernels
We consider the impulse dynamical system: = F (x1 , x2 ) = (+x2 + X −a (x1 , x2 ) X x1 , −x1 + n+1 n (xn+1 , x ) = Φ(x , x ) = (x − 1 , xn2 ) 1 2 1 1 2
X −a X x2 )
(14)
where the set of constrains is K = {(x1 , x2 ) ∈ [−c, c]2 , (x1 , x2 ) ≥ ε}, c > a > ε.
Fig. 2. Structure of the strict hybrid kernel whenever C ∩ V iabF (K) is ∅ or not. The continuous dynamic describes a process for which the circle C(0, a) centered at O and of radius a is the V iabF (K) which is an unstable limit circle: in the lack of impulse any solution starting out this circle leaves K in finite time. On figure 2 up right, the reset set is a segment C = {X = (x1 , x2 ) | x1 = 1, x2 ∈ [−1, 1]} which does not encounter V iabF (K). The viability kernel V iabF (K) is contained in the hybrid kernel and periodic runs are contained
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in V iabF (K). On figure 2 up left, C = {X = (x1 , x2 ) | x1 = 0.5, x2 ∈ [−1, 1]} encounters V iabF (K). This changes the structure of the hybrid kernel: periodic runs are no more contained in V iabF (K) but they exist somewhere else. On figure 2 down, the reset set is a pair of segments Cα = {X = (x1 , x2 ) such that (x1 + 0.4x2 = 1 or x1 − 0.4x2 = 1) and x2 ∈ [−1, 1]}. The hybrid kernel is shown down right and the strict hybrid kernel is shown down left. We can see that the topological structure of hybrid kernels may become very complex.
3
Capture Basin and Minimal Time Function for Target Problem with Impulses
Let T be a closed target to be reached in finite time satisfying T ∩ C = ∅. The Capture Basin of T is the domain of the minimal time-to-reach function and the epigraph of this function is the viability kernel of an extended dynamic (see [7, Cardaliaguet, Quincampoix & Saint-Pierre]). This holds true in presence of impulses. In the same context, in [9] E. Cr¨ uck has obtained similar results allowing the development of numerical schemes approximating the Minimal Time function for impulse dynamics. Let us define F (x) if x ∈ /T FT (x) := co(F (x) ∪ {0}) if x ∈ T The set-valued map FT is Marchaud and we deduce from Theorem 2.10: Theorem 3.1 (Viable Capture Basin) Let us assume (3) and (4). Then Hyb(FT ,Φ) (K) is the largest closed set of initial points for which there exists at least one hybrid solution viable in K which either reaches T in finite time or remains in the hybrid kernel for F of K\T : Hyb(F,Φ) (K\T ). Let us introduce the extended impulse dynamic (Ψ, Ξ) with constraint K × R+ n+1
(x
(x , z ) ∈ ΨT (x, z) := FT (x) × {−1} (or FT (x) × [0, 1] if x ∈ T ) , z n+1 ) ∈ Ξ(xn , z n ) := (Φ(xn ), z n )
Theorem 3.2 (Minimal Time Function) Let us assume (3) and (4). Then Hyb(ΨT ,Ξ) (K × R+ ) is the largest closed set of initial points (x0 , z0 ) for which there exists at least one hybrid solution x(t) viable in K until it reaches T in a finite time τ ≤ z0 . Moreover, V (x0 ) = min{z0 | (x0 , z0 ) ∈ Hyb(ΨT ,Ξ) (K × R+ )} is the Minimal Time-to-reach function which is the smallest positive lower semicontinuous supersolution of the following HJB equation d (15) ∀ x ∈ Dom(V )\C, max < f (x, u), − V (x) > −1 = 0 u∈ U dx The proof is also a consequence of Theorem 2.10 and is similar to the proof given in [7] for control problems. The domain of V is the K-Viable Capture Basin of C for (F, Φ). Example - Consider the so-called Zermelo swimmer problem studied in [7]. We look for the minimal time function. Let us introduce an impulse dynamic
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allowing jumps from some points to other ones given by the following impulse dynamical system: (x , y , z ) = (b − ay + c u, c v, −1) ∈ F (X) f f (16) (xn+1 , y n+1 , z n+1 ) = (xn − 2 + cr u, y n + cr v, z n ) ∈ Φ(X) where control (u, v) ∈ B(0, 1), K = [−6, 6] × (−5, 5] × R+ and C = {(i, j), i ∈ {−6, −4, −2, 0, 2, 4}, j ∈ {−2, −1, 1, 2}}
Fig. 3. The impulse dynamic and the level curves of the Minimal Time function. Dynamical system and target are shown on figure 3 left. The continuous evolution corresponds to the down arrows pointing to a ball of radius cf , the impulse evolution correspond to the up arrows pointing to a ball of radius cr . The constraint set K is the square area. The reset set is the collection of dark points. The level curves of the minimal time function are represented on the right. The knowledge of this (approximated) function, solution to the HJB equation, allows to determine the optimal (approximated) synthesis: two (approximated) optimal viable trajectories are computed, the one starting from A and reaching the target in a minimal time is superimposed on the right.
Annex:
Proofs of Propositions 2.8 and 2.9
Proof of Proposition 2.8 ——- The usual way to prove inclusion (12) is to construct a suitable sequence of equicontinuous piecewise linear approximations and, thanks to Ascoli-Arzela and the Convergence Theorems, to check right properties of the limit. However, for impulse systems, the equicontinuity property of approximated solutions fails at reset times. For this reason, to bypass the lack of equicontinuity of the hybrid solution, we introduced the expanded hybrid solution. e ρ∗ (resp. x0 ∈ Kρ∗ ) and let xdρ = (x0 , x1 , ..., xn , ...) be a discrete Let ρ be fixed, x0 ∈ K solution to (8) i) (resp. (8) ii)) starting from x0 and viable in K obtained by choosing ϕn ∈ Fρ (xn ) (we omit when it is not necessary - but do not forget - the dependency of all items with respect to ρ). With xdρ we associate the continuous piecewise linear function ϑ → xeρ (ϑ) given by xeρ (ϑ) =
xn + (ϑ − ϑn )ϕn ∀ϑ ∈ [ϑn , ϑn + ρ[ n n n n+1 n n 1 − (x + ρϕ )] ∀ϑ ∈ [ϑn + ρ, ϑn+1 [ x + ρϕ + δ (ϑ − (ϑ + ρ))[x
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The second line in the definition of xeρ (ϑ) is effective only if ϑn+1 = ϑn + ρ + δ, that is when a reset occurs at step n. If not, ϑn+1 = ϑn + ρ. Thanks to such a transformation we build a continuous, piecewise linear and differentiable solution except when ϑ = ϑn and ϑ = ϑn + ρ where it admits only right derivative. For all n and for all ϑ ∈]ϑn , ϑn+1 [ we have dxeρ (ϑ) = dϑ
ϕn if ϑ ∈]ϑn , ϑn + ρ[ n+1 n n 1 (x − (x + ρϕ )) if ϑ ∈]ϑn + ρ, ϑn+1 [ δ
In order to distinguish the evolution governed by the continuous dynamic and the δ evolution governed by the impulse dynamic we choose δ = Minf . Also, on interval +1 n n ]ϑ , ϑ + ρ[, the velocity is upper bounded by M + 1 and, in the second case, if xn+1 = (xn + ρϕn ), that is to say when a reset occurs, the velocity is constant and lower bounded by M + 1. If d(xn+1 , xn ) ≤ ρM , that is if r n+1 = r n , from (4) we have for all ϑ ∈]ϑn , ϑn+1 [: dxeρ (ϑ) ∈ Fρ (xn ) ⊂ F (xeρ (ϑ)) + φ(ρ)B ⊂ B(0, M + φ(ρ)) dϑ If d(xn+1 , xn ) > ρM , that is if r n+1 = r n + 1, the previous property holds true for all ϑ ∈]ϑn , ϑn + ρ[ and, from (3) we have for all ϑ ∈]ϑn + ρ, ϑn + ρ + δ[: dxeρ (ϑ) = y n ∈ Φh (xn + ρϕn ) − (xn + ρϕn ) dϑ dxe
and so, on this interval, the derivative is constant and satisfies : (M + 1) ≤ dϑρ (ϑ) ≤ δ . Thus we have (M + 1) δsup inf dxeρ δsup (ϑ) ≤ (M + 1) sup dϑ δinf ϑ≥0 The family of functions ϑ → xeρ (ϑ) is equicontinuous. Since K is compact, the family is uniformly bounded and from Ascoli-Arzela theorem, it is relatively compact. There exists a subsequence denoted (ρk )k converging to 0 such that xeρk (·) converges uniformly to a function xe (t) when k → ∞: xeρk (ϑ) → xe (ϑ), uniformly ∀ϑ ≥ 0
(17)
d e Moreover, from Alaoglu theorem, the sequence of derivatives dϑ xρ (·) is bounded in ∞ −bϑ L (O, ∞; X, e dϑ), it is included in a weakly compact subset of L1 (O, ∞; X, e−bϑ dϑ). There exists a subsequence still denoted (ρk )k such that xeρk converges weakly to the derivative of xe (·) when k → ∞:
dxeρk dxe (ϑ) * (ϑ), weakly a.e. ϑ ≥ 0 dϑ dϑ
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Lemma 3.3 For all ϑ0 , one of the two following properties holds true: e e i) ∃η0 > 0 such that ∀ϑ ∈]ϑ0 , ϑ0 + η0 ], dx dϑ (ϑ) ≤ M and x (ϑ) ∈ K e e ii) ∃ϑ ∈ [ϑ0 − δ, ϑ0 ], ∃y ∈ Φ(x (ϑ)) − x (ϑ) s.t. ∀ϑ ∈]ϑ, ϑ + δ[, xe (ϑ) = xe (ϑ) + ϑ−ϑ δ y. e
In the first case xe (·) is a locally viable solution to differential inclusion dx dϑ ∈ e 1 F (xe ) defined on a right neighborhood of ϑ0 . In the second case dx (ϑ) = dϑ δ y on ]ϑ, ϑ + δ[.
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Proof of Lemma ——- Let ϑ0 be strictly positive. Consider the following property (Pϑ ) ∃η0 > 0, ∃k0 ∈ N, (ϑ ∈ [ϑ0 , ϑ0 + η0 ] and k ≥ k0 ⇒ and the alternative: “(Pϑ ) is true or false”:
dxe ρ
k
dϑ
(ϑ) ≤ M + ϕ(ρk ))
dxe ρ
a) If (Pϑ ) is true then, by construction dϑk (ϑ) ∈ F (xeρk (ϑ)) + ϕ(ρk )B. Let ϑ be fixed and k going to infinity. Since F is upper semi continuous with convex compact e (·) values, Convergence Theorem ([1, Aubin] Thm. 2.2.4) implies that the weak limit dx dϑ satisfies dxe dxe (ϑ) ∈ F (xe (ϑ)) and (ϑ) ≤ M for almost all ϑ ∈ [ϑ0 , ϑ0 + η0 ]. dϑ dϑ b) If (Pϑ ) is false, its contrapositive holds true and since the norm of the derivative
dxe ρ (ϑ) dϑ
is by construction either lower to M + ϕ(ρ) or greater or equal to M + 1 we dxeρk (ϑ) > M + 1. have: ∀η > 0, ∀k, ∃ϑ ∈ [ϑ0 , ϑ0 + η], ∃k ≥ k s.t. dϑ Then there exists a step nk such that i) ii) iii) iv)
ϑnk + ρk ≤ ϑ < ϑnk + ρk + δ xeρk (ϑnk ) = xnk xeρk (ϑnk + ρk ) = xnk + ρk ϕnk ∈ Ch xeρk (ϑnk + ρk + δ) = xnk +1 ∈ Φ(xeρk (ϑnk + ρk ))
Consequently, if we set y ρk =
xnk +1 − (xnk + ρk ϕnk ) we have δ
dxeρk Φh (xnk + ρk ϕnk ) − (xnk + ρk ϕnk ) (ϑ) = y ρk ∈ . dϑ δ ϑ − (ϑnk + ρk ) y ρk δ Let η → 0 and k → ∞. Since ϑ ∈ [ϑ0 , ϑ0 + η] we have from i)
and xeρk (ϑ) = xnk + ρk ϕnk +
ϑnk + ρk ≤ ϑ ≤ ϑ0 + η and ϑ0 ≤ ϑ < ϑnk + ρk + δ so that, since there exists a subsequence of ϑnk which converges to ϑ, since functions xeρ (·) converge uniformly and since Φ is closed, we have i) ii) iii) iv)
ϑ ≤ ϑ0 ≤ ϑ ≤ ϑ + δ xe (ϑ) ∈ C xe (ϑ + δ) ∈ Φ(xe (ϑ)) e e (ϑ) xe (ϑ) = xe (ϑ) + (ϑ − ϑ) x (ϑ+δ)−x . δ
Φ(xe (ϑ)) − xe (ϑ) dxe 1 (ϑ) = y ∈ , ∀ ϑ ∈ [ϑ, ϑ + δ[ Moreover, since the sedϑ δ δ quence xnk is contained in K, the limit solution xe (ϑ) is contained in K except the open segment joining any point in C to its reset position which membership to K is not required.
So we have
Reconstruction of the hybrid solution. It remains to prove that the limit solution xe (ϑ) is an expanded hybrid solution associated with the initial impulse system (F, Φ, K). We can assume that x0 ∈ / C. If not we will start, if it exists, from the first reset position which does not belong to C. From the limit solution xe (·) we recover the
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hybrid solution by a change of variable. For this task, we consider the sequence ϑi defined from ϑ0 = 0 as follows: dxe (ϑ) ≥ M + 1} ϑ1 = inf{ϑ > ϑ0 such that xe (ϑ) ∈ C and dϑ+ and ϑ1 = +∞ if it does not exists a time ϑ > 0 where xe (ϑ) belongs to C and is reset at Φ(xe (ϑ)). If not dxe ϑi = inf{ϑ ≥ ϑi−1 + δ such that xe (ϑ) ∈ C} and (ϑ) ≥ M + 1} dϑ+ and ϑi = +∞ if there is no further time ϑ > ϑi−1 + δ where xe (ϑ) belongs to C. With any ϑ > 0 we associate the index i(ϑ) = {i | ϑi ≤ ϑ < ϑi+1 }, we set t = ϑ − i(ϑ)δ and, for all index i, ti = ϑi − iδ. Then the map t → x∗ (t) := xe (t + iδ), for all t ∈ [ti , ti+1 [ satisfies x∗ (ti ) = xe (ϑi ) for all i. When ϑ spans [0, +∞), t spans some interval [0, T ). The upper bound T may be finite or infinite. It is finite when, after a finite number of switches between continuous and impulse evolutions, the solution becomes purely impulsive jumping from C ∩ K to C ∩ K. We have now to check that the solution x∗ (·) is an hybrid solution associated with − → satisfying the definition 2.1. Indeed, for all i > 0, on each interval ]ti , ti+1 [, a run x∗ the general convergence theorem (see [1, Aubin], [16, Saint-Pierre]) implies that the solution x∗ (·) remains in K and satisfies the differential inclusion dx∗ (t) ∈ F (x∗ (t)) for almost all t ∈]ti , ti+1 [. dt Let us set τi := ti+1 − ti , x∗i := x∗ (ti ) and let us define the motive x∗i (·) by = x∗ (ti + τ ) on the interval [0, τi ]. At any time ti < +∞ the solution is reset from the position x∗ (ti ) = x∗i (τi ) ∈ C to the position x∗i+1 ∈ Φ(x∗ (ti )). Then, from this position starts the solution x∗ (·) which is viable in K forever or until it reaches anew C at position x∗ (ti+1 ) where it is reset at a position belonging to Φ(x∗ (ti+1 )). So that the corresponding run is precisely − → = {(τi , x∗ , x∗ (·)}. x∗ i i e ρ∗ or We have proved that, for any ρ and for any sequence of elements x0ρ ∈ K 0 ∗ 0 xρ ∈ Kρ , when ρ → 0, there exists a subsequence converging to some element x and a − → starting from x0 at time t = 0 which is a viable run associated with the impulse run x∗ system (F, Φ, K). x∗i (τ )
This ends the proof of Proposition 2.8. Proof of Proposition 2.9 ——- Let x0 ∈ Hyb(F,Φ) (K) be an arbitrary initial → position. There exists at least one viable run − x = {(τi , xi , xi (·)} with x0 = x0 solution to the impluse system (F, Φ, K). Let us denote x(·) an hybrid solution starting from x0 and remaining in K. Let ρ > 0 be a given time step Psatisfying condition (9). With any τi we associate τ ) + 1 and ni = ij=1 νj with ν0 = n0 = 0. νi = Integer( i−1 ρ Let us define the sequence (xρn )n as follows: for all n ∈ N we set - in := {i ∈ I such that ni ≤ n < ni+1 }, - xρ,n = xin (n − nin ), with xin (0) = xin and rρ,n = in We shall prove that the sequence (xρ,n )n∈N is a solution to the discrete system (8) and conclude that x0 belongs to any Kρ∗ . Indeed we have
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− xρ,n ≤ ρM , then for all t ∈ [nρ, (n + 1)ρ], x (t) ∈ F (x(t)). Since - either xρ,n+1 P xρ,n+1 ∈ xρ,n + ρ0 F (x(ρn + τ ))dτ , we deduce from (4-ii)) that xρ,n+1 ∈ xρ,n + ρFρ (xρ,n ) ⊂ Sρ (xρ,n ) - or xρ,n+1 − xρ,n > ρM , then from (3), there exists t∗ ∈ [nρ, (n + 1)ρ[ such that x(t∗ ) ∈ C and a reset occurs at this time. This necessarily implies that xρ,n ∈ Ch since, from (5), h ≥ M ρ and that xρ,n+1 ∈ Φh (xρ,n ) ⊂ Sρ (xρ,n ).
References [1] AUBIN J.-P. (1991) Viability Theory Birkh¨ auser, Boston, Basel, Berlin [2] AUBIN J.-P. (1999) Impulse Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, University of California at Berkeley [3] AUBIN J.-P. & CELLINA A. (1984) Differential inclusions, SpringerVelag, Grundlehren der math. Wiss. # 264 [4] AUBIN J.-P. & FRANKOWSKA H. (1990) Set-Valued Analysis, [5] BENSOUSSAN A. & LIONS J.-L. (1984) Impulse control and quasivariational inequalities, Gauthier-Villars [6] BENSOUSSAN A. & MENALDI (1997) Hybrid Control and Dynamic Programming, Dynamics of Continuous, Discrete and Impulse Systems, 3, 395442 [7] CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1997) Optimal times for constrained non-linear control problems without local controllability Applied Mathematics & Optimization, 36:21-42 [8] CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1999) Set-valued numerical methods for optimal control and differential games, In Stochastic and differential games. Theory and numerical methods, Annals of the International Society of Dynamical Games, 177-247 Birkh¨ auser ¨ [9] CRUCK E. (2001) Probl`emes de cible sous contrainte d’´etat pour des syst`emes non lin´eaires avec sauts d’´etat , C.R. Acad. Sci. Paris, t. 333, S´erie I, p. 403-408. [10] DORDAN O. (1995) Analyse qualitative, Masson [11] DOYEN L. & SAINT-PIERRE P. (1997) Scale of viability and minimal time of crisis, Set-Valued Analysis, 5, 227-246 [12] FRANKOWSKA H. (1991) Lower semicontinuous solutions to HamiltonJacobi-Bellman equations, Proceedings of 30th CDC Conference, IEEE, Brighton. [13] LEITMANN G., (1979) Guaranteed Asymptotic Stability for a Class of Uncertain Linear Dynamical Systems. Journ. of Optimization Theory and Applic., 27(1). [14] QUINCAMPOIX M. (1992) Differential inclusions and target problems, SIAM J. Control and Optimization, 30, 324-335 [15] QUINCAMPOIX M. & VELIOV V. (1998) Viability with a target: theory and applications, in Applications of math. in engineering, 47-54, Heron Press [16] SAINT-PIERRE P., (1994) Approximation of the Viability Kernel, Applied Mathematics & Optimization, 29 (1994), 187-209. [17] SAINT-PIERRE P., (1996) Equilibria and stability in setvalued analysis: a viabili-ty approach. Topology in nonlin. Analysis, Banach Center Pub. 35,243-255 Warsaw
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Patrick Saint-Pierre [18] SEUBE N. (1995) Robust Stabilization of Uncertain Systems. Journal of Math. Analysis and Applications, 452-466. [19] SHAFT (van der) A. & SCHUMACHER H. (1999) An introduction to hybrid dynamical systems, Springer-Verlag, Lecture Notes in Control, 251 [20] SASTRY S., (1999) Non Linear Systems. Analysis, Stability and Control, Springer-Verlag. [21] VAANDRAGER F.W. & VAN SCHUPPEN J.H. (1999) Hybryd Systems: Computation and ControL Vol 1569 of LNCS. Springer-Verlag. Berlin. [22] ZABCZYK J. (1973) Optimal Control by means of switching Studia Mathematica, 65,161-171
Abstract. We investigate, for constrained controlled systems with impulse, the subset of initial positions contained in a set K from which starts at least one run viable in K - the hybrid viability kernel - eventually until it reaches a given closed target in finite time - the hybrid capture basin. We define a constructive algorithm which approximates this set. The knowledge of this set is essential for control problem since it provides viable hybrid feed-backs and viable runs. We apply this method for approximating the Minimal Time-to-reach Function in the presence of both constraints and impulses. Two examples are presented, the first deals with a dynamical system revealing the complexity of the structure of hybrid kernels, the second deals with a Minimal Time problem with impulses.
1
Introduction
We consider a dynamical impulse system describing the evolution of a state variable x ∈ X = Rn which, in response to some events, may switch between a continuous evolution and an impulse evolution. Switches are triggered when the state reaches a closed set C. During some periods the state is governed by a continuous evolution until it reaches some state x ∈ C where a reset to a new position x+ ∈ Φ(x) occurs. We will distinguish between solution which, when reaching C, either may be reset or keep going the continuous evolution, and strict solution which, when reaching C, is necessarily reset. Research efforts are devoted to the study of such systems mainly in extending the mathematical tools of the Control Theory (see [22, Zabczyk], [5, Bensoussan & Lions] and in developing methods for analysis, verification or control design for non linear hybrid systems (see for instance [20, Sastry], [19, Shaft & Schumacher], [21, Vaandrager & Van Schuppen], [2, Aubin]). On the other hand the study of controlled system with state constraints has been widely developed this last decade using the main concepts of Set-Valued Analysis (see [3, Aubin & Cellina], [4, Aubin & Frankowska]) and Numerical SetValued Analysis (see [16, Saint-Pierre]) which can be regarded as a “Tool Box” for viability theory and numerical viability (viability, equilibria, stability, reachability, controllability, (see [1, Aubin], [14, Quincampoix], [17, Saint-Pierre]), optimal control (minimal time-to-reach problem, crisis time function, infinite C.J. Tomlin and M.R. Greenstreet (Eds.): HSCC 2002, LNCS 2289, pp. 378–392, 2002. c Springer-Verlag Berlin Heidelberg 2002
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horizon optimal control (see [7, Cardaliaguet, Quincampoix & Saint-Pierre], [11, Doyen & Saint-Pierre]), differential games (discriminating and leadership kernels, conditional and guaranteed strategies, minimal hitting time function, robustness, qualitative analysis (see [13, Leitmann], [18, Seube], [8, Cardaliaguet, Quincampoix & Saint-Pierre], [10, Dordan]) and their applications to mathematical economics, finance, demography or biology as well as to enginery and automatics when taking into account constraints is unescapable. The scope of this paper is to bridge these two fields in order to take into account explicit or hidden constraints that may occur when studying hybrid systems and to explore how numerical schemes can be implemented. Let us recall that the Viability Kernel Algorithm originally designed for computing viability kernels has been widen to approximate the smallest lower semicontinuous subsolution of the Hamilton-Jacobi-Belmann equation (see [12, Frankowska], [7, Cardaliaguet, Quincampoix & Saint-Pierre]). We prove that viability techniques can be extended with minor restrictions to hybrid systems first for computing the Hybrid Kernel which broadens the concept of Viability Kernel, second for computing the Minimal Time-to-reach function as example of Value Function.
2
Hybrid Viability Kernels
2.1
Definitions: Run, Hybrid, Strict Hybrid Solutions
X that we assume to be Mar1. Let us consider a set-valued map F : X chaud1 . The continuous evolution is described by the differential inclusion x (t) ∈ F (x(t)), for almost all t ∈ R+
(1)
We denote by SFc (x0 ) the set of all absolutely continuous solutions of (1) starting from x0 at time t0 = 0. We denote by θC (x(·)) the first time t ≥ 0 when the solution x(·) reaches C. 2. Let us consider a set-valued map Φ defined on a closed set C = Dom(Φ) with compact values and closed graph. The impulse evolution is described by the recursive inclusion (2) xn+1 ∈ Φ(xn ) 3. We denote by SΦd (x0 ) the set of all discrete solutions {x0 , x1 , ..., xk } of (2) starting from x0 ∈ C. If finite, k is the first index such that xk ∈ / C. Definition 2.1 We call run of an impulse system (F, Φ) a sequence of elements − → x (·) := {(τi , xi , xi (·))}i∈I ∈ (R+ ×X ×C(0, ∞; X))N, where τi is the ith cadence, xi is the ith reinitialization, with x0 = x0 , and xi (·) ∈ SFc (xi ) is the ith motive which is an almost continuous solution to (1) starting from xi at time 0 until time τi . 1
Upper semicontinuous with non empty convex compact values and linear growth.
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We denote by S(F,Φ) (x0 ) the set of runs starting from x0 . The set of indexes I = {0, 1, ..., n} ⊂ N can be finite (n < +∞) or infinite (n = +∞), satisfying ∀i < n, ∃xi (·) ∈ SFc (xi ) such that xi (τi ) ∈ C, xi+1 ∈ Φ(xi (τi )) If τi = 0, xi+1 ∈ Φ(xi ) and then xi (·) is defined on an interval of length 0. n We set T = i=0 τi . Let us introduce a virtual lapse of time δ > 0, sequences (ti )i∈I and (ϑi )i∈I given by t0 = ϑ0 = 0, ti+1 = ti +τi and ϑi+1 = ϑi +τi +δ. We have ϑi+1 = ti + iδ. With any t ∈ [0, T ] we associate it satisfying tit ≤ t < tit +1 and with any ϑ ∈ R we associate iϑ satisfying ϑiϑ ≤ ϑ < ϑiϑ +1 . → Definition 2.2 With a run − x (·) we associate the expanded hybrid solution as e the map ϑ → x (ϑ) defined by iϑ τi e x (ϑ) = x0 + xi (τ )dτ + αi (xi+1 − xi (τi )) i=0
0
where the motive xi (τ ) is a solution to (1) starting from xi at time 0 until time τi , τi = min(τi , ϑ − ϑi ) and αi = max(0, min(1, ϑ − ϑi − τi )). We call hybrid → solution associated with a run − x (·) the map t → x(t) given by x(t) = xe (t + it δ). The expanded hybrid and the hybrid solutions are two parametrized representations of the same run. However, under slight assumption, there exists a one-to-one correspondence between expanded hybrid solutions and runs which fails when considering plain hybrid solutions. For studying impulse systems, that we call strict, when the solution is necessarily reset every times the trajectory reaches C to some position x∗ belonging to Φ(x∗ ), we need to introduce the following Definition 2.3 We call strict hybrid solution any hybrid solution such that ∀i ∈ I, xi (τi ) ∈ C and ∀τ ∈ [0, τi [, xi (τ ) ∈ / C. e e (F,Φ) (x0 ))) the set of extended (resp. strict) hybrid (x0 ) (resp. S We note S(F,Φ) solutions starting from x0 .
Let K be a closed set. We search the largest domain of initial positions from which starts at least one run remaining forever in K. We can assume without loss of generality that C = K ∩ Φ−1 (K) and we denote by S(F,Φ,K) (x0 ) the set → of runs − x (·) starting from x0 and viable in K. An impulse constrained system is characterized by the triple (F, Φ, K). The notion of hybrid kernel has been introduced in [2, Aubin]. It is characterized in terms of capture basin when K\C is a repeller2 . 2
We recall that a set D is a repeller if all solutions leave D in a finite time. The capture basin of a set C under constraint K for the dynamic F , denoted CaptF (C, K), is the set of initial position x0 from which there exists a solution x(·) ∈ SFc (x0 ) remaining in K until it reaches C in finite time.
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Definition 2.4 The hybrid (resp. strict hybrid) kernel of K for the impulse system (F, Φ, K) is the largest closed subset of initial states belonging to K from which starts at least one (resp. strict) hybrid viable solution. We denote this set Hyb(F,Φ) (K) (resp. Hyb (F,Φ) (K)). We have Hyb (F,Φ) (K) ⊂ Hyb(F,Φ) (K). We are going to prove that the hybrid kernel is closed and that hybrid and strict hybrid kernels can be approximated by a sequence of discrete viability kernels associated with suitable discrete systems. Assumptions and Approximated Impulse Systems We assume that • C is compact and Φ : C 0 < δinf ≤ inf
X is upper semi-continuous with compact values: inf d(x, y) ≤ sup sup d(x, y) ≤ δsup
x∈C y∈Φ(x)
(3)
x∈C y∈Φ(x)
so that Φ has no fix point and its graph is closed. We set ∀x ∈ / C, Φ(x) = ∅. • K is a compact set and F is a Marchaud map satisfying sup sup y ≤ M . x∈K y∈F (x)
This assumption implies that F is closed. As usual in the context of set-valued numerical analysis we need to consider “good” approximations Fρ of F in the sense that Graph(Fρ ) remains in a not too large neighborhood of F i) Graph(F ρ ) ⊂ Graph(F ) + φ(ρ)BX×X ii) F (x ) ⊂ Fρ (x)
(4)
x ∈B(x,ρM)
where BX×X denotes the unit ball of X × X and φ goes to zero when ρ. Let be h > 0 and set Ch := C + hB. With the set-valued map Φ we associate Φh (x) :=
Φ( x)
x e∈B(x,h)∩C
Proposition 2.5 If Φ is closed (its graph is closed) on C ×X, then Φh is closed. Proof —– Let (xn , y n ) ∈ Graph(Φh ) be a sequence converging to (x∗ , y ∗ ). From the definition of Φh , ∀n, ∃x en ∈ B(xn , h) ∩ C such that yn ∈ Φ(xen ). Since xen belongs to a compact subset and since C is closed, there exists a subsequence x en converging to ∗ ∗ ∗ x e ∈ B(x , h) ∩ C. Since Φ is closed on C, we have y ∈ Φ(xe) ⊂ Φh (x ) and consequently (x∗ , y ∗ ) ∈ Graph(Φh ).
Consistency of numerical schemes subordinates h to ρ in such a way that any element of x + ρFρ (x) sufficiently close to C belongs to Ch . We can choose3 h = ρ(M + φ(ρ)). 3
For computing viability kernels we refer to [16, Saint-Pierre].
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The Viability Kernel Algorithm
−−→ The discrete viability kernel of K for G denoted V iabG (K) is the largest closed subset of K of initial position from which there exists at least one sequence solution to the discrete dynamical system remaining in K: xn+1 ∈ G(xn ), ∀n ∈ N
(6)
The viability kernel algorithm consists in the construction of a decreasing sequence of subsets K n recursively defined by K 0 = K, and K n+1 = K n ∩ {x | G(x) ∩ K n = ∅} = K n ∩ G−1 (K n )
(7)
−−→ which converges to V iabG (K) (cf. [16, Saint-Pierre]): Proposition 2.6 Let G : X X be an upper semi-continuous and compact −−→ set valued map and K a compact set. Then Limn→∞ K n = V iabG (K). Let Fρ −−→ satisfying (4) and set Gρ = Id + ρFρ , then Limρ→0 V iabGρ (K) = V iabF (K) 2.3
Approximation Schemes for Impulse Systems
Let us fix a time step ρ and h = ρ(M + φ(ρ)). We replace the derivative x (t) of n+1 n x at time t by the difference x ρ−x where xn stands for x(nρ) with x0 = x(0).
n+1 We consider the set x = xn + ρϕn | ϕn ∈ Fρ (xn ) of successors of xn . We set if y ∈ / Ch y y if y ∈ / Ch Sρ (y) = and S ρ (y) = y ∪ Φh (y) if y ∈ ∂Ch y ∪ Φh (y) if y ∈ Ch Φh (y) if y ∈ Int(Ch ) Since Φh is closed, the graphs of set-valued maps Sρ and S ρ are also by construction closed. We consider the discrete dynamical systems i) xn+1 ∈ Gρ (xn ) = {Sρ (xn + ρϕ)| ϕ ∈ Fρ (xn )} ρ (xn ) = S ρ (xn + ρϕ)| ϕ ∈ Fρ (xn ) ii) xn+1 ∈ G
(8)
Proposition 2.7 If F and Φ are closed and if F is compact valued, then for all ρ are closed. ρ > 0, Gρ and G The proof is similar to the proof of Proposition 2.5. Let us choose ρ such that (9) 2M ρ < δinf We add to the discrete dynamical systems (8) i) and ii) two variables which evaluate a discrete time tn and the number of resets that took place until tn : n r if d(xn+1 , xn ) ≤ ρM tn+1 = tn + ρ and rn+1 = n r + 1 if d(xn+1 , xn ) ≥ δinf Thanks to assumption (3), evaluation of the distance between two successive states allows to distinguish between continuous and impulse evolutions and consequently to know whenever xn is or isn’t in a neighborhood of C.
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2.4
Impulse Viability Algorithm p and K p by 0 = K 0 = K. We define sequences K Let be K ρ ρ ρ ρ ρp+1 = {x ∈ K ρp | G ρ (x) ∩ K ρp = ∅}, Kρp+1 = {x ∈ Kρp | Gρ (x) ∩ Kρp = ∅} (10) K In other words, x ∈ Kρp+1 if and only if x ∈ Kρp and ∃ϕ ∈ Fρ (x) such that Sρ (x + ρϕ) ∩ Kρp = ∅ that is to say that x + ρϕ ∈ Sρ−1 (Kρp ) and (10) becomes: ρp+1 := (Id + ρFρ )−1 (S ρ−1 (K ρp )) ∩ K ρp , Kρp+1 := (Id + ρFρ )−1 (Sρ−1 (Kρp )) ∩ Kρp K ρp and Kρp Since the graphs of S ρ , Sρ and Id+ρFρ are closed and K is compact, K p p are compact. Decreasing sequences Kρ and Kρ converge in the sense of Painlev´e−→ −−→ ρ∗ = − Kuratowski to limit sets K ViabGeρ (K) and Kρ∗ = ViabGρ (K) satisfying ∗ = (1 + ρFρ )−1 (S −1 (K ∗ )) and K ∗ = (1 + ρFρ )−1 (S −1 (K ∗ )) K ρ ρ ρ ρ ρ ρ 2.5
(11)
Approximation of the Hybrid Viability Kernels
We have to prove that the limit Limρ→0 Kρ∗ exists and is equal to Hyb(F,Φ) (K) and that it remains true, under restrictive assumptions for strict impulse systems. This derives from the following properties which proofs are annexed. Proposition 2.8 Under assumptions (3) and (4), we have ∗ ⊂ Hyb(F,Φ) (K) Limsupρ→0 Kρ∗ ⊂ Hyb(F,Φ) (K) and Limsupρ→0 K ρ
(12)
and Proposition 2.9 Let us assume (3) and (4). Then Hyb(F,Φ) (K) ⊂ Kρ∗ . from which we deduce our main result Theorem 2.10 (Hybrid Kernel) Let us assume (3) and (4). Then Hyb(F,Φ) (K) = Limρ→0 Kρ∗ which is closed. Case of strict impulse system The difference between hybrid and strict hybrid solutions arises from that any strict solution cannot encounter C without being reset whereas hybrid solution may follow a continuous path even if it encounters C. The strict hybrid kernel is not necessarily closed as shown in the following example. Exemple Let us consider the impulse dynamical system given by F (x, y) = (−1, 0) Φ(x, y) = {x + 1} × {x + y + 1} (13) C = {(x, y) such that sup(|x|, |y|) = 1 and ((x ≤ 0) or (y ≥ 0))} K = [−2, 2] × [−2, 2] It is easy to check (see figure 1) that the strict hybrid kernel is not closed. We aim to give a sufficient condition implying the convergence of discrete ∗ to the strict hybrid kernel strict hybrid kernels K ρ r Let be ∂C := {x ∈ ∂C such that ∃px , T∂C (x) = {y | < px , y >= 0}}, the regular part of the boundary of C.
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Fig. 1. Closed hybrid and non closed strict hybrid kernels.
Proposition 2.11 (Strict Hybrid Kernel) Let us assume (3), (4), that F is Lipschitz on a neighborhood of C and that the two following properties hold true: (A0) ∀x ∈ ∂C r , F (x) ∩ T∂C (x) = ∅, (A1) ∀x ∈ ∂C\∂C r any solution of (1) starting from x leaves K in finite time without reaching C. ∗ Then Hyb (F,Φ) (K) = Limρ→0 Kρ and Hyb(F,Φ) (K) is closed.
e ρ∗ a sequence converging to some element x0 . From PropoProof —— Let be x0ρ ∈ K → x = {(τi , xi , xi (·)}i∈I sition 2.8, x0 ∈ Hyb(F,Φ) (K) and so there exists a viable run − →ρ = {(τρi , xρi , starting from x0 solution of system (F, Φ, K) and there exist viable runs − x xρi (·)}i∈I starting from x0ρ such that the sequence of associated expanded strict solutions xeρ (·) converge uniformly to xe (·) when ρ → 0. g (F,Φ)(K). Then, there exists j ∈ I such that xj (0) = xj , Assume that x0 ∈ / Hyb xj (τj ) ∈ C if τj < ∞ and ∃τbj ∈]0, τj [ such that xj (τbj ) ∈ ∂C. • If xj (τbj ) ∈ ∂C r , from assumption (A0) there exist η > 0 and γ > 0 such that < pxj (τbj ) , xj (τbj − η) − xj (τbj ) >< −γ and < pxj (τbj ) , xj (τbj + η) − xj (τbj ) >> γ. Using a continuity argument, since F is Lipschitz, ∃ρb such that ∀ρ ≤ ρb: ∃τbρj ∈ [τbj − η, τbj + η] ⊂]0, τρj [, xρj (τbρj ) ∈ ∂C →ρ is a x which is impossible since xρ (·) is the hybrid solution associated with the run − strict hybrid solution. • If xj (τbj ) ∈ ∂C\∂C r , then from assumption (A1) the hybrid solution x(·) associ→ ated with the viable run − x would leave K in finite time which is impossible.
We can replace assumption (A1) by a global assumption of emptyness of the viability kernel of K\C under F (see [2, Aubin]). Corollary 2.12 Let us assume (3) and (4). If K is a repeller for F and if no solution starting from C and immediately leaving C reaches C then ∗ Hyb (F,Φ) (K) = Limρ→0 Kρ .
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Proof ——- Since no solution starting and immediately leaving from C reaches C and, since K is a repeller for F , all solutions leave K in a finite time. So that any hybrid viable solution which encounters C at some position x∗ must necessarily be reset from x∗ to a position belonging to Φ(x∗ ) for maintaining viability.
2.6
Exemple of Numerical Computation of Hybrid Kernels
We consider the impulse dynamical system: = F (x1 , x2 ) = (+x2 + X −a (x1 , x2 ) X x1 , −x1 + n+1 n (xn+1 , x ) = Φ(x , x ) = (x − 1 , xn2 ) 1 2 1 1 2
X −a X x2 )
(14)
where the set of constrains is K = {(x1 , x2 ) ∈ [−c, c]2 , (x1 , x2 ) ≥ ε}, c > a > ε.
Fig. 2. Structure of the strict hybrid kernel whenever C ∩ V iabF (K) is ∅ or not. The continuous dynamic describes a process for which the circle C(0, a) centered at O and of radius a is the V iabF (K) which is an unstable limit circle: in the lack of impulse any solution starting out this circle leaves K in finite time. On figure 2 up right, the reset set is a segment C = {X = (x1 , x2 ) | x1 = 1, x2 ∈ [−1, 1]} which does not encounter V iabF (K). The viability kernel V iabF (K) is contained in the hybrid kernel and periodic runs are contained
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in V iabF (K). On figure 2 up left, C = {X = (x1 , x2 ) | x1 = 0.5, x2 ∈ [−1, 1]} encounters V iabF (K). This changes the structure of the hybrid kernel: periodic runs are no more contained in V iabF (K) but they exist somewhere else. On figure 2 down, the reset set is a pair of segments Cα = {X = (x1 , x2 ) such that (x1 + 0.4x2 = 1 or x1 − 0.4x2 = 1) and x2 ∈ [−1, 1]}. The hybrid kernel is shown down right and the strict hybrid kernel is shown down left. We can see that the topological structure of hybrid kernels may become very complex.
3
Capture Basin and Minimal Time Function for Target Problem with Impulses
Let T be a closed target to be reached in finite time satisfying T ∩ C = ∅. The Capture Basin of T is the domain of the minimal time-to-reach function and the epigraph of this function is the viability kernel of an extended dynamic (see [7, Cardaliaguet, Quincampoix & Saint-Pierre]). This holds true in presence of impulses. In the same context, in [9] E. Cr¨ uck has obtained similar results allowing the development of numerical schemes approximating the Minimal Time function for impulse dynamics. Let us define F (x) if x ∈ /T FT (x) := co(F (x) ∪ {0}) if x ∈ T The set-valued map FT is Marchaud and we deduce from Theorem 2.10: Theorem 3.1 (Viable Capture Basin) Let us assume (3) and (4). Then Hyb(FT ,Φ) (K) is the largest closed set of initial points for which there exists at least one hybrid solution viable in K which either reaches T in finite time or remains in the hybrid kernel for F of K\T : Hyb(F,Φ) (K\T ). Let us introduce the extended impulse dynamic (Ψ, Ξ) with constraint K × R+ n+1
(x
(x , z ) ∈ ΨT (x, z) := FT (x) × {−1} (or FT (x) × [0, 1] if x ∈ T ) , z n+1 ) ∈ Ξ(xn , z n ) := (Φ(xn ), z n )
Theorem 3.2 (Minimal Time Function) Let us assume (3) and (4). Then Hyb(ΨT ,Ξ) (K × R+ ) is the largest closed set of initial points (x0 , z0 ) for which there exists at least one hybrid solution x(t) viable in K until it reaches T in a finite time τ ≤ z0 . Moreover, V (x0 ) = min{z0 | (x0 , z0 ) ∈ Hyb(ΨT ,Ξ) (K × R+ )} is the Minimal Time-to-reach function which is the smallest positive lower semicontinuous supersolution of the following HJB equation d (15) ∀ x ∈ Dom(V )\C, max < f (x, u), − V (x) > −1 = 0 u∈ U dx The proof is also a consequence of Theorem 2.10 and is similar to the proof given in [7] for control problems. The domain of V is the K-Viable Capture Basin of C for (F, Φ). Example - Consider the so-called Zermelo swimmer problem studied in [7]. We look for the minimal time function. Let us introduce an impulse dynamic
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allowing jumps from some points to other ones given by the following impulse dynamical system: (x , y , z ) = (b − ay + c u, c v, −1) ∈ F (X) f f (16) (xn+1 , y n+1 , z n+1 ) = (xn − 2 + cr u, y n + cr v, z n ) ∈ Φ(X) where control (u, v) ∈ B(0, 1), K = [−6, 6] × (−5, 5] × R+ and C = {(i, j), i ∈ {−6, −4, −2, 0, 2, 4}, j ∈ {−2, −1, 1, 2}}
Fig. 3. The impulse dynamic and the level curves of the Minimal Time function. Dynamical system and target are shown on figure 3 left. The continuous evolution corresponds to the down arrows pointing to a ball of radius cf , the impulse evolution correspond to the up arrows pointing to a ball of radius cr . The constraint set K is the square area. The reset set is the collection of dark points. The level curves of the minimal time function are represented on the right. The knowledge of this (approximated) function, solution to the HJB equation, allows to determine the optimal (approximated) synthesis: two (approximated) optimal viable trajectories are computed, the one starting from A and reaching the target in a minimal time is superimposed on the right.
Annex:
Proofs of Propositions 2.8 and 2.9
Proof of Proposition 2.8 ——- The usual way to prove inclusion (12) is to construct a suitable sequence of equicontinuous piecewise linear approximations and, thanks to Ascoli-Arzela and the Convergence Theorems, to check right properties of the limit. However, for impulse systems, the equicontinuity property of approximated solutions fails at reset times. For this reason, to bypass the lack of equicontinuity of the hybrid solution, we introduced the expanded hybrid solution. e ρ∗ (resp. x0 ∈ Kρ∗ ) and let xdρ = (x0 , x1 , ..., xn , ...) be a discrete Let ρ be fixed, x0 ∈ K solution to (8) i) (resp. (8) ii)) starting from x0 and viable in K obtained by choosing ϕn ∈ Fρ (xn ) (we omit when it is not necessary - but do not forget - the dependency of all items with respect to ρ). With xdρ we associate the continuous piecewise linear function ϑ → xeρ (ϑ) given by xeρ (ϑ) =
xn + (ϑ − ϑn )ϕn ∀ϑ ∈ [ϑn , ϑn + ρ[ n n n n+1 n n 1 − (x + ρϕ )] ∀ϑ ∈ [ϑn + ρ, ϑn+1 [ x + ρϕ + δ (ϑ − (ϑ + ρ))[x
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The second line in the definition of xeρ (ϑ) is effective only if ϑn+1 = ϑn + ρ + δ, that is when a reset occurs at step n. If not, ϑn+1 = ϑn + ρ. Thanks to such a transformation we build a continuous, piecewise linear and differentiable solution except when ϑ = ϑn and ϑ = ϑn + ρ where it admits only right derivative. For all n and for all ϑ ∈]ϑn , ϑn+1 [ we have dxeρ (ϑ) = dϑ
ϕn if ϑ ∈]ϑn , ϑn + ρ[ n+1 n n 1 (x − (x + ρϕ )) if ϑ ∈]ϑn + ρ, ϑn+1 [ δ
In order to distinguish the evolution governed by the continuous dynamic and the δ evolution governed by the impulse dynamic we choose δ = Minf . Also, on interval +1 n n ]ϑ , ϑ + ρ[, the velocity is upper bounded by M + 1 and, in the second case, if xn+1 = (xn + ρϕn ), that is to say when a reset occurs, the velocity is constant and lower bounded by M + 1. If d(xn+1 , xn ) ≤ ρM , that is if r n+1 = r n , from (4) we have for all ϑ ∈]ϑn , ϑn+1 [: dxeρ (ϑ) ∈ Fρ (xn ) ⊂ F (xeρ (ϑ)) + φ(ρ)B ⊂ B(0, M + φ(ρ)) dϑ If d(xn+1 , xn ) > ρM , that is if r n+1 = r n + 1, the previous property holds true for all ϑ ∈]ϑn , ϑn + ρ[ and, from (3) we have for all ϑ ∈]ϑn + ρ, ϑn + ρ + δ[: dxeρ (ϑ) = y n ∈ Φh (xn + ρϕn ) − (xn + ρϕn ) dϑ dxe
and so, on this interval, the derivative is constant and satisfies : (M + 1) ≤ dϑρ (ϑ) ≤ δ . Thus we have (M + 1) δsup inf dxeρ δsup (ϑ) ≤ (M + 1) sup dϑ δinf ϑ≥0 The family of functions ϑ → xeρ (ϑ) is equicontinuous. Since K is compact, the family is uniformly bounded and from Ascoli-Arzela theorem, it is relatively compact. There exists a subsequence denoted (ρk )k converging to 0 such that xeρk (·) converges uniformly to a function xe (t) when k → ∞: xeρk (ϑ) → xe (ϑ), uniformly ∀ϑ ≥ 0
(17)
d e Moreover, from Alaoglu theorem, the sequence of derivatives dϑ xρ (·) is bounded in ∞ −bϑ L (O, ∞; X, e dϑ), it is included in a weakly compact subset of L1 (O, ∞; X, e−bϑ dϑ). There exists a subsequence still denoted (ρk )k such that xeρk converges weakly to the derivative of xe (·) when k → ∞:
dxeρk dxe (ϑ) * (ϑ), weakly a.e. ϑ ≥ 0 dϑ dϑ
(18)
Lemma 3.3 For all ϑ0 , one of the two following properties holds true: e e i) ∃η0 > 0 such that ∀ϑ ∈]ϑ0 , ϑ0 + η0 ], dx dϑ (ϑ) ≤ M and x (ϑ) ∈ K e e ii) ∃ϑ ∈ [ϑ0 − δ, ϑ0 ], ∃y ∈ Φ(x (ϑ)) − x (ϑ) s.t. ∀ϑ ∈]ϑ, ϑ + δ[, xe (ϑ) = xe (ϑ) + ϑ−ϑ δ y. e
In the first case xe (·) is a locally viable solution to differential inclusion dx dϑ ∈ e 1 F (xe ) defined on a right neighborhood of ϑ0 . In the second case dx (ϑ) = dϑ δ y on ]ϑ, ϑ + δ[.
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Proof of Lemma ——- Let ϑ0 be strictly positive. Consider the following property (Pϑ ) ∃η0 > 0, ∃k0 ∈ N, (ϑ ∈ [ϑ0 , ϑ0 + η0 ] and k ≥ k0 ⇒ and the alternative: “(Pϑ ) is true or false”:
dxe ρ
k
dϑ
(ϑ) ≤ M + ϕ(ρk ))
dxe ρ
a) If (Pϑ ) is true then, by construction dϑk (ϑ) ∈ F (xeρk (ϑ)) + ϕ(ρk )B. Let ϑ be fixed and k going to infinity. Since F is upper semi continuous with convex compact e (·) values, Convergence Theorem ([1, Aubin] Thm. 2.2.4) implies that the weak limit dx dϑ satisfies dxe dxe (ϑ) ∈ F (xe (ϑ)) and (ϑ) ≤ M for almost all ϑ ∈ [ϑ0 , ϑ0 + η0 ]. dϑ dϑ b) If (Pϑ ) is false, its contrapositive holds true and since the norm of the derivative
dxe ρ (ϑ) dϑ
is by construction either lower to M + ϕ(ρ) or greater or equal to M + 1 we dxeρk (ϑ) > M + 1. have: ∀η > 0, ∀k, ∃ϑ ∈ [ϑ0 , ϑ0 + η], ∃k ≥ k s.t. dϑ Then there exists a step nk such that i) ii) iii) iv)
ϑnk + ρk ≤ ϑ < ϑnk + ρk + δ xeρk (ϑnk ) = xnk xeρk (ϑnk + ρk ) = xnk + ρk ϕnk ∈ Ch xeρk (ϑnk + ρk + δ) = xnk +1 ∈ Φ(xeρk (ϑnk + ρk ))
Consequently, if we set y ρk =
xnk +1 − (xnk + ρk ϕnk ) we have δ
dxeρk Φh (xnk + ρk ϕnk ) − (xnk + ρk ϕnk ) (ϑ) = y ρk ∈ . dϑ δ ϑ − (ϑnk + ρk ) y ρk δ Let η → 0 and k → ∞. Since ϑ ∈ [ϑ0 , ϑ0 + η] we have from i)
and xeρk (ϑ) = xnk + ρk ϕnk +
ϑnk + ρk ≤ ϑ ≤ ϑ0 + η and ϑ0 ≤ ϑ < ϑnk + ρk + δ so that, since there exists a subsequence of ϑnk which converges to ϑ, since functions xeρ (·) converge uniformly and since Φ is closed, we have i) ii) iii) iv)
ϑ ≤ ϑ0 ≤ ϑ ≤ ϑ + δ xe (ϑ) ∈ C xe (ϑ + δ) ∈ Φ(xe (ϑ)) e e (ϑ) xe (ϑ) = xe (ϑ) + (ϑ − ϑ) x (ϑ+δ)−x . δ
Φ(xe (ϑ)) − xe (ϑ) dxe 1 (ϑ) = y ∈ , ∀ ϑ ∈ [ϑ, ϑ + δ[ Moreover, since the sedϑ δ δ quence xnk is contained in K, the limit solution xe (ϑ) is contained in K except the open segment joining any point in C to its reset position which membership to K is not required.
So we have
Reconstruction of the hybrid solution. It remains to prove that the limit solution xe (ϑ) is an expanded hybrid solution associated with the initial impulse system (F, Φ, K). We can assume that x0 ∈ / C. If not we will start, if it exists, from the first reset position which does not belong to C. From the limit solution xe (·) we recover the
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hybrid solution by a change of variable. For this task, we consider the sequence ϑi defined from ϑ0 = 0 as follows: dxe (ϑ) ≥ M + 1} ϑ1 = inf{ϑ > ϑ0 such that xe (ϑ) ∈ C and dϑ+ and ϑ1 = +∞ if it does not exists a time ϑ > 0 where xe (ϑ) belongs to C and is reset at Φ(xe (ϑ)). If not dxe ϑi = inf{ϑ ≥ ϑi−1 + δ such that xe (ϑ) ∈ C} and (ϑ) ≥ M + 1} dϑ+ and ϑi = +∞ if there is no further time ϑ > ϑi−1 + δ where xe (ϑ) belongs to C. With any ϑ > 0 we associate the index i(ϑ) = {i | ϑi ≤ ϑ < ϑi+1 }, we set t = ϑ − i(ϑ)δ and, for all index i, ti = ϑi − iδ. Then the map t → x∗ (t) := xe (t + iδ), for all t ∈ [ti , ti+1 [ satisfies x∗ (ti ) = xe (ϑi ) for all i. When ϑ spans [0, +∞), t spans some interval [0, T ). The upper bound T may be finite or infinite. It is finite when, after a finite number of switches between continuous and impulse evolutions, the solution becomes purely impulsive jumping from C ∩ K to C ∩ K. We have now to check that the solution x∗ (·) is an hybrid solution associated with − → satisfying the definition 2.1. Indeed, for all i > 0, on each interval ]ti , ti+1 [, a run x∗ the general convergence theorem (see [1, Aubin], [16, Saint-Pierre]) implies that the solution x∗ (·) remains in K and satisfies the differential inclusion dx∗ (t) ∈ F (x∗ (t)) for almost all t ∈]ti , ti+1 [. dt Let us set τi := ti+1 − ti , x∗i := x∗ (ti ) and let us define the motive x∗i (·) by = x∗ (ti + τ ) on the interval [0, τi ]. At any time ti < +∞ the solution is reset from the position x∗ (ti ) = x∗i (τi ) ∈ C to the position x∗i+1 ∈ Φ(x∗ (ti )). Then, from this position starts the solution x∗ (·) which is viable in K forever or until it reaches anew C at position x∗ (ti+1 ) where it is reset at a position belonging to Φ(x∗ (ti+1 )). So that the corresponding run is precisely − → = {(τi , x∗ , x∗ (·)}. x∗ i i e ρ∗ or We have proved that, for any ρ and for any sequence of elements x0ρ ∈ K 0 ∗ 0 xρ ∈ Kρ , when ρ → 0, there exists a subsequence converging to some element x and a − → starting from x0 at time t = 0 which is a viable run associated with the impulse run x∗ system (F, Φ, K). x∗i (τ )
This ends the proof of Proposition 2.8. Proof of Proposition 2.9 ——- Let x0 ∈ Hyb(F,Φ) (K) be an arbitrary initial → position. There exists at least one viable run − x = {(τi , xi , xi (·)} with x0 = x0 solution to the impluse system (F, Φ, K). Let us denote x(·) an hybrid solution starting from x0 and remaining in K. Let ρ > 0 be a given time step Psatisfying condition (9). With any τi we associate τ ) + 1 and ni = ij=1 νj with ν0 = n0 = 0. νi = Integer( i−1 ρ Let us define the sequence (xρn )n as follows: for all n ∈ N we set - in := {i ∈ I such that ni ≤ n < ni+1 }, - xρ,n = xin (n − nin ), with xin (0) = xin and rρ,n = in We shall prove that the sequence (xρ,n )n∈N is a solution to the discrete system (8) and conclude that x0 belongs to any Kρ∗ . Indeed we have
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− xρ,n ≤ ρM , then for all t ∈ [nρ, (n + 1)ρ], x (t) ∈ F (x(t)). Since - either xρ,n+1 P xρ,n+1 ∈ xρ,n + ρ0 F (x(ρn + τ ))dτ , we deduce from (4-ii)) that xρ,n+1 ∈ xρ,n + ρFρ (xρ,n ) ⊂ Sρ (xρ,n ) - or xρ,n+1 − xρ,n > ρM , then from (3), there exists t∗ ∈ [nρ, (n + 1)ρ[ such that x(t∗ ) ∈ C and a reset occurs at this time. This necessarily implies that xρ,n ∈ Ch since, from (5), h ≥ M ρ and that xρ,n+1 ∈ Φh (xρ,n ) ⊂ Sρ (xρ,n ).
References [1] AUBIN J.-P. (1991) Viability Theory Birkh¨ auser, Boston, Basel, Berlin [2] AUBIN J.-P. (1999) Impulse Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, University of California at Berkeley [3] AUBIN J.-P. & CELLINA A. (1984) Differential inclusions, SpringerVelag, Grundlehren der math. Wiss. # 264 [4] AUBIN J.-P. & FRANKOWSKA H. (1990) Set-Valued Analysis, [5] BENSOUSSAN A. & LIONS J.-L. (1984) Impulse control and quasivariational inequalities, Gauthier-Villars [6] BENSOUSSAN A. & MENALDI (1997) Hybrid Control and Dynamic Programming, Dynamics of Continuous, Discrete and Impulse Systems, 3, 395442 [7] CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1997) Optimal times for constrained non-linear control problems without local controllability Applied Mathematics & Optimization, 36:21-42 [8] CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1999) Set-valued numerical methods for optimal control and differential games, In Stochastic and differential games. Theory and numerical methods, Annals of the International Society of Dynamical Games, 177-247 Birkh¨ auser ¨ [9] CRUCK E. (2001) Probl`emes de cible sous contrainte d’´etat pour des syst`emes non lin´eaires avec sauts d’´etat , C.R. Acad. Sci. Paris, t. 333, S´erie I, p. 403-408. [10] DORDAN O. (1995) Analyse qualitative, Masson [11] DOYEN L. & SAINT-PIERRE P. (1997) Scale of viability and minimal time of crisis, Set-Valued Analysis, 5, 227-246 [12] FRANKOWSKA H. (1991) Lower semicontinuous solutions to HamiltonJacobi-Bellman equations, Proceedings of 30th CDC Conference, IEEE, Brighton. [13] LEITMANN G., (1979) Guaranteed Asymptotic Stability for a Class of Uncertain Linear Dynamical Systems. Journ. of Optimization Theory and Applic., 27(1). [14] QUINCAMPOIX M. (1992) Differential inclusions and target problems, SIAM J. Control and Optimization, 30, 324-335 [15] QUINCAMPOIX M. & VELIOV V. (1998) Viability with a target: theory and applications, in Applications of math. in engineering, 47-54, Heron Press [16] SAINT-PIERRE P., (1994) Approximation of the Viability Kernel, Applied Mathematics & Optimization, 29 (1994), 187-209. [17] SAINT-PIERRE P., (1996) Equilibria and stability in setvalued analysis: a viabili-ty approach. Topology in nonlin. Analysis, Banach Center Pub. 35,243-255 Warsaw
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Patrick Saint-Pierre [18] SEUBE N. (1995) Robust Stabilization of Uncertain Systems. Journal of Math. Analysis and Applications, 452-466. [19] SHAFT (van der) A. & SCHUMACHER H. (1999) An introduction to hybrid dynamical systems, Springer-Verlag, Lecture Notes in Control, 251 [20] SASTRY S., (1999) Non Linear Systems. Analysis, Stability and Control, Springer-Verlag. [21] VAANDRAGER F.W. & VAN SCHUPPEN J.H. (1999) Hybryd Systems: Computation and ControL Vol 1569 of LNCS. Springer-Verlag. Berlin. [22] ZABCZYK J. (1973) Optimal Control by means of switching Studia Mathematica, 65,161-171
