Optimality Conditions for Optimal Impulsive Control Problems with Multipoint State Constraints Olga N. Samsonyuk Matrosov Institute for System Dynamics and Control Theory of SB RAS, Lermontova str., 134, 664033 Irkutsk, Russia samsonyuk.olga@gmail.com Abstract This paper deals with optimal impulsive control problems whose states are functions of bounded variation and impulsive controls are regular vector measures. The problem under consideration has multipoint state constraints. Examples of such problems may be found in mechanics, quantum electronics, robotics, ecology, economics, etc. Sufficient global optimality conditions involving certain sets of Lyapunov type functions are proposed. These Lyapunov type functions are strongly monotone solutions of the corresponding Hamilton-Jacobi inequalities. 1 Introduction This paper concerns an optimal impulsive control problem in which the control system is an extension of the control system ( ) ( ) ẋ(t) = f t, x(t), V (t), u(t) + G t, x(t), V (t) v(t), (1) u(t) ∈ U, v(t) ∈ K a.e. on T, (2) where T = [a, b] is a fixed time interval, U is a compact set in Rr , K is a convex closed cone in Rm , x(·) ∈ W 1,1 (T, Rn ), u(·) ∈ L∞ (T, Rr ), v(·) ∈ L∞ (T, Rm ). Function V (·) is the total variation on [a, t] for the function ∫ t . ∑ m . t → w(t) = v(τ )dτ , that is, V (t) = var wi (·). The symbol “a.e.” signifies “almost everywhere with respect a [a,t] i=1 to the Lebesgue measure, L”. In general, optimization problems over the control system (1), (2) do not have solutions in the class of absolutely continuous trajectories and Lebesgue measurable controls. This is explained by the fact that the right-hand side of (1) is pointwise unbounded. Thereby minimizing sequences of trajectories may pointwise tend to discontinuous functions. By closing the set of solutions of (1), (2) in the weak∗ topology in the space of functions of bounded variation, we obtain an impulsive control system which can be formally described as follows ( ) ( ) dx(t) = f t, x(t), V (t), u(t) dt + G t, x(t), V (t) µ, (3) Copyright ⃝ c by the paper’s authors. Copying permitted for private and academic purposes. In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org 497 u(t) ∈ U a.e. on T, µ(B) ∈ K ∀ B ∈ BT . (4) Here, µ is a K-valued bounded Borel measure on T , x(·) is a function of bounded variation, BT is the( set )of all Borel subsets of T, and V (·) is a nonnegative nondecreasing function such that V (b) ≥ |µ| [a, b] , where the measure |µ| is the total variation of µ. Let us note that any interpretation of (3), (4) as a measure-driven differential equation cannot provide a concept of solution with well-posedness properties [Bressan & Rampazzo, 1988], [Miller, 1996], [Miller & Rubinovich, 2003], [Motta & Rampazzo, 1995], [Motta & Rampazzo, 1996], [Motta & Rampazzo, 1996], [Sesekin & Zavalishchin, 1997]. There exist many solu- tions corresponding to given u(·), µ, and an initial point x(a). This is due to the fact that we do not assume any commutativity property of the vector fields generated by the columns of G. Namely, generally the Lie brackets [Gi , Gj ], i, j = 1, m, do not vanish identically. To overcome this drawback we extend the notion of impulsive control to a pair π(µ) consisting of µ and an additional component γ(µ) defined below. Such γ(µ) character- izes a way of approximation of µ by some sequences of L-absolutely continuous measures µk = vk (t)dt, where vk : T → K. Throughout this paper we assume that the following conditions are satisfied. H1. The functions f (t, x, V, u), G(t, x, V ) are continuous; for any compact set Q ⊂ Rn there exist constants L1Q , L2Q > 0 such that |f (t, x1 , V, u) − f (t, x2 , V, u)| ≤ L1Q |x1 − x2 |, |G(t, x1 , V ) − G(t, x2 , V )| ≤ L2Q |x1 − x2 | whenever (t, x1 , V, u), (t, x2 , V, u) ∈ T × Q × R+ × U. Moreover, there exist constants c1 , c2 > 0 such that |f (t, x, V, u)| ≤ c1 (1 + |x|), |G(t, x, V )| ≤ c2 (1 + |x|) whenever (t, x, V, u) ∈ T × Rn × R+ × U. Here, R+ = {y ∈ R | y ≥ 0}; | · | denotes a vector norm or a consistent matrix norm. H2. The set f (t, x, V, U ) is a convex set for every (t, x, V ) ∈ T × Rn × R+ . ∑ m Denote K1 = {v ∈ K | ||v|| = 1}, where ||v|| = |vj |, and let co A be the convex hull of a set A. Let µ be a j=1 bounded Borel measure on T whose values are from K. Given µ, we denote by µc , |µc |, and Sd (µ) the continuous component in the Lebesgue decomposition of the measure µ, total variation of the measure µc , and the set on . which the discrete component ( )of µ is concentrated, that is, Sd (µ) = {s ∈ T | µ({s}) ̸= 0}, respectively. Let π(µ) be a pair µ, γ(µ) whose components satisfy the following conditions: i) µ is a K-valued bounded Borel measure on T , ii) γ(µ) is a set {ds , ωs (·)}s∈S such that (a) S ⊇ Sd (µ) and S is at most countable subset of T , (b) for every s ∈ S , ds ∈ R+ and ωs is L-measurable function [0, ds ] → co K1 such that ∫ ds ( ) ( ) ds ≥ ||µ {s} ||, ωs (τ )dτ = µ {s} , 0 ∑ (c) ds < ∞. s∈S The element π(µ) is called an impulsive control. We denote by W(T, K) the set of all impulsive controls π(µ) satisfying conditions i), ii). The second control u(·) has to be regarded as an ordinary control from L∞ (T, Rr ) such that u(t) ∈ U a.e. t ∈ T. Given controls u(·), π(µ) and an initial condition x(a), consider the system of differential equations with the measure ( ) ( ) ∑ ( ) dx(t) = f t, x(t), V (t), u(t) dt + G t, x(t), V (t) µc + zs (ds ) − x(s−) δ(t − s), (5) s∈S, s≤t 498 ∑ ( ) dV (t) = |µc | + zV s (ds ) − V (s−) δ(t − s), V (a) = 0, t ∈ T, (6) s∈S, s≤t dzs (τ ) ( ) = G s, zs (τ ), zV s (τ ) ωs (τ ), zs (0) = x(s−), (7) dτ dzV s (τ ) = 1, zV s (0) = V (s−), τ ∈ [0, ds ], s ∈ S. (8) dτ ( ) Let x(·), V (·) be a solution of (5)–(8). Then x(·), V (·) are functions of bounded variation. Without loss of generality, suppose that x(·) and V (·) are right continuous on (a, b]. From (6), (8) it follows that ( ) ∑ V (t) = |µc | [a, t] + ds , t ∈ (a, b]. s≤t, s∈S Now we consider the impulsive control system (D) ( ) ( ) dx(t) = f t, x(t), V (t), u(t) dt + G t, x(t), V (t) π(µ), (9) ( ) u(t) ∈ U a.e. on T, π(µ) ∈ W T, K . (10) ( ) Let u(·) and π(µ) satisfy (10). The tuple σ = XV , u(·), π(µ) is said to be an impulsive process of (D) if XV is a set-valued function acting from T to comp(Rn+1 ) such that {( )} i) ∀ t ∈ T /S XV (t) = x(t), V (t) , {( ) } ii) for every s ∈ S XV (s) = zs (τ ), zV s (τ ) | τ ∈ [0, ds ] . ( ) ( ) Here, functions x(·), V (·) , zs (·), zV s (·) satisfy (5)–(8) with some initial condition x(a), comp(Rn+1 ) is the set of non-empty compact subsets from Rn+1 . Set by definition {( )} {( )} XV (t−) = x(t−), V (t−) ∀ t ∈ (a, b], XV (a−) = x(a), 0 , {( )} {( )} XV (t+) = x(t+), V (t+) ∀ t ∈ [a, b), XV (b+) = x(b), V (b) . ( ) Denote by Σ the set of all impulsive processes σ = XV , u(·), π(µ) of (D). Let us briefly comment on the relation between (D) and the corresponding conventional system (1), (2). Given XV , define its graph on T to be . { } graph XV = (t, x, V ) | t ∈ T, (x, V ) ∈ XV (t) . T Let A, B ∈ comp(R n+1 ). Denote by d(A, B) the Hausdorff ( distance) between A and B. { Lemma 1 [Samsonyuk,} 2015]. 1) Let σ = X V , u(·), π(µ) ∈ Σ. Then, there exists a sequence xk (·), Vk (·), uk (·), vk (·) such that i) for every k, the functions xk (·), Vk (·), uk (·), vk (·) satisfy (1), (2); ii) ( ( )) d graph XV , graph xk , Vk → 0, k → ∞. (11) T T { } 2) Let xk (·), Vk (·), uk (·), vk (·) be a sequence of functions such that i) sup ||vk (·)||L1 < ∞; k ii) for every k, the functions xk (·), Vk (·), uk (·), vk (·) satisfy (1), (2); iii) {xk (a)} is bounded. ( ) { } Then, there exist σ = XV , u(·), π(µ) ∈ Σ and a subsequence xkj (·), Vkj (·), ukj (·), vkj (·) such that ( ( )) d graph XV , graph xkj , Vkj → 0, j → ∞. T T 499 2 Statement of the Problem Let θ( = (θ0 , . . . , θk)) be a vector of fixed points of time such that a ≤ θ0 < · · · < θk ≤ b, k < ∞. Given σ = XV , u(·), π(µ) ∈ Σ, define the vector qσ that is composed of the one-sided limits of XV at the points θj , j = 1, k; i.e., . ( ) qσ = {XV (θj −)}j=0,k , {XV (θj +)}j=0,k . Let us consider the optimal impulsive control problem P (θ) with multipoint state constraints minimize J(σ) = l(qσ ) subject to σ ∈ Σ, qσ ∈ C. Here, C is a closed set in Rd(qσ ) , where d(qσ ) is the dimension of qσ , l : Rd(qσ ) → R is a continuous function. A process σ ∈ Σ is said to be a feasible process of P (θ) if qσ ∈ C. 3 Preliminaries The aim of this section is to recall some facts about Lyapunov type functions and their monotone property relative to impulsive control problems with trajectories of bounded variation [Samsonyuk, 2010], [Dykhta & Samsonyuk, 2015]. Let φ be a continuous function T × Rn × R+ 7→ R. Let (tα , xα ) ∈ T × Rn , Vα ≥ 0. Define the sets T , TVα (tα , xα ), and Qφ (tα , xα , Vα ) as follows { ( ) } { } T = XV | ∃ σ = XV , u(·), π(µ) ∈ Σ , TVα (tα , xα ) = XV ∈ T | XV (tα −) = (xα , Vα ) , { } Qφ (tα , xα , Vα ) = (t, x, V ) ∈ T × Rn × R+ | φ(t, x, V ) ≤ φ(tα , xα , Vα ) . Definition 1. Function φ is strongly decreasing relative to (D) if for any (tα , xα ) ∈ T × Rn , Vα ≥ 0 and for any XV ∈ TVα (tα , xα ) the inclusion graph XV ⊂ Qφ (tα , xα , Vα ) [tα ,b] holds. This monotone property may be formulated in term of decreasing φ along of all solutions of (D). Indeed, φ is said to be decreasing along XV if for any (t1 , x1 , V1 ) and (t2 , x2 , V2 ), where (x1 , V1 ) ∈ XV (t1 ) and (x2 , V2 ) ∈ XV (t2 ) such that t1 < t2 or V1 < V2 , the inequality φ(t1 , x1 , V1 ) ≥ φ(t2 , x2 , V2 ) is fulfilled. Then φ is strongly decreasing if φ decreases along any XV ∈ T . Let us note that such functions are usual called Lyapunov type functions. For optimal impulsive control problem with multipoint state constraints it is natural to consider compound Lyapunov type functions which will be defined below. Let ρ be some partition of T by points {t0 , t1 , . . . , tN } such that a = t0 < t1 < . . . < tN = b. Denote ∆i = (ti−1 , ti ), i = 1, N , and let T∆i be the restriction of T to ∆i . First, given ∆i , we consider the system of proximal Hamilton–Jacobi inequalities pt + H0 (t, x, px ) ≤ 0 ∀ p = (pt , px , pV ) ∈ ∂P φ(t, x, V ), ∀ (t, x, V ) ∈ (ti−1 , ti ) × Rn × [0, +∞), (12) pV + H1 (t, x, px ) ≤ 0 ∀ p = (pt , px , pV ) ∈ ∂P φ(t, x, V ), ∀ (t, x, V ) ∈ [ti−1 , ti ] × Rn × (0, +∞). (13) Here, H0 (t, x, ψ) = max⟨ψ, f (t, x, u)⟩, H1 (t, x, ψ) = max ⟨ψ, G(t, x)ω⟩, the set ∂P φ(t, x, V ) is the proximal u∈U ω∈K1 subdifferential of φ at the point (t, x, V ). Let us recall [Clarke et al., 1998], [Vinter, 2000] that a vector p ∈ Rd(y) is called a proximal subgradient of a function y → φ(y) at a point y if there exist a neighborhood Q of the point y and a constant c > 0 such that φ(z) ≥ φ(y) + ⟨p, z − y⟩ − c|z − y|2 ∀ z ∈ Q. This inequality implies that locally (in a neighborhood of y) φ has a quadratic lower support function at the point y with gradient p at this point. The proximal subdifferential ∂P φ(y) consists of all subgradients. In the case ∂P φ(y) is empty, the respective proximal inequalities are assumed to hold automatically at the point y. 500 Note that ∂P φ(y) ⊂ {∇φ(y)} if φ is differentiable; moreover, the last inclusion turns into the equality if φ is twice continuously differentiable at y. Let Φ∆i be the set of all continuous solutions of (12), (13). Then Φ∆i consists of strongly decreasing on ∆i functions. Namely, if φ ∈ Φ∆i , then φ decreases along any XV ∈ T∆i . Next, for every tj , j = 0, N , we consider the so-called limiting system ( ) z ′ (τ ) = G tj , z(τ ), zV (τ ) ω(τ ), zV′ (τ ) = 1, ω(τ ) ∈ co K1 a.e. τ ≥ 0 (14) with Lebesgue measurable controls ω(·). Let LTtj be the set of solutions of (14). Let us recall that a continuous function ξ(z, zV ) strongly decreases relative to (14) if ξ(z, zV ) is a solution of the proximal Hamilton–Jacobi inequality pzV + H1 (tj , z, pz ) ≤ 0 ∀ (pz , pzV ) ∈ ∂P ξ(z, zV ), ∀ (z, zV ) ∈ Rn × (0, +∞) (15) (for more details see [Clarke et al., 1998]). Denote by Ξtj the set of all continuous solutions of (15). ( ) Definition 2. The set {φi }i=1,N , {ξj }j=0,N , where φi ∈ Φ∆i , i = 1, N , ξj ∈ Ξtj , j = 0, N , is called a compound strongly decreasing function. { } Given Φ∗∆i ⊂ Φ∆i , i = 1, N , and Ξ∗tj ⊂ Ξtj , j = 0, N , we define Φ∗ρ to be Φ∗ρ = {Φ∗∆i }i=1,N , {Ξ∗tj }j=0,N . Then Φ∗ρ is called a set of compound strongly decreasing functions. 4 Main Results In this section ( sufficient global ) optimality conditions for problem P (θ) will be formulated. Let σ = XV , u(·), π(µ) ∈ Σ be a feasible process of P (θ) and let ρ = {t0 , . . . , tN }, where a = t0 < t1 < . . . < tN = b, be some partition of T including all θj , j = 0, k, i.e., ρ ⊇ {θ0 , . . . , θk }. Denote by I the set {j ∈ {0, . . . , N } | tj ∈ {θ0 , . . . , θr }} . In what follows we use the notation ( ) j . j . . { j j } qσ0 = XV (tj −), qσ1 = XV (tj +), j = 0, N , qσ,ρ = qσ0 , qσ1 j=0,N . (16) Given ρ, define the sets {( ) } X∆i = (xi−1 , Vi−1 ), (xi , Vi ) ∃ XV ∈ T : XV (ti−1 +) = (xi−1 , Vi−1 ), XV (ti −) = (xi , Vi ) , i = 1, N , { ( ) } ( ) ∃ z(·), zV (·) ∈ LTtj : Ztj = (z0 , zV 0 ), (z1 , zV 1 ) . , j = 0, N , z(0) = z0 , z(d) = z1 , zV (d) − zV (0) = d, d = zV 1 − zV 0 { ({ } ) ( i−1 i ) ( ) } Rρ = q = q0j , q1j j=0,N q1 , q0 ∈ X∆i , i = 1, N , q0j , q1j ∈ ZEtj , j = 0, N . Let us note that the set Rρ consists of points connected by trajectories of (D). This set may be interpreted as a reachable set corresponding to ρ. It is easy to see that, for any σ ∈ Σ and any ρ, the corresponding qσ,ρ belongs to Rρ . And the { contrary, for any q ∈ R }ρ there exists σ ∈ Σ such that qσ,ρ = q. Let Φ∗ρ = {Φ∗∆i }i=1,N , {Ξ∗tj }j=0,N be an arbitrary set of compound strongly decreasing functions. Define the sets ∩ {( ) } A[Φ∗∆i ] = q1 , q0 | q0 , q1 ∈ Rn × R+ , φ(ti , q0 ) − φ(ti−1 , q1 ) ≤ 0 , i = 1, N , φ∈Φ∗ ∆ i ∩ {( ) } LA[Ξ∗tj ] = q0 , q1 | q0 , q1 ∈ Rn × R+ , ξ(q1 ) − ξ(q0 ) ≤ 0 , j = 0, N , ξ∈Ξ∗ t j { ({ } ) ( i−1 i ) ( ) } A[Φ∗ρ ] = q = q0j , q1j j=0,N q1 , q0 ∈ A[Φ∗∆i ], i = 1, N , q0j , q1j ∈ LA[Ξ∗tj ], j = 0, N . By using the strong decreasing property of functions from Φ∗∆i , i = 1, N , and Ξ∗tj , j = 0, N , one can readily obtain that A[Φ∗∆i ], i = 1, N , and LA[Ξ∗tj ], j = 0, N , give outer estimations for X∆i , i = 1, N and Ztj , j = 0, N , respectively; i.e., X∆i ⊆ A[Φ∗∆i ], i = 1, N , Ztj ⊆ LA[Ξ∗tj ], j = 0, N . (17) 501 From (17) the next result follows. Lemma 2. Let ρ be a partition of T and Φ∗ρ be an arbitrary set of compound strongly decreasing functions. Then Rρ ⊆ A[Φ∗ρ ]. Now let us (formulate) sufficient optimality conditions. Denote by AP (θ) the finite-dimensional optimization problem . ({ } ) . ({ } ) l(qI ) → min; qI ∈ C, q ∈ A[Φ∗ρ ], where q = q0j , q1j j=0,N , qI = q0j , q1j j∈I . ( ) Let σ = X V , u, π(µ) be an examining process of P (θ) and q σ be the corresponding vector of the one-sided limits of X V . Let the vector q σ,ρ be defined by (16). ( ) The set Φ∗ρ is said to be resolving for σ if the vector q σ,ρ is a global minimum point in the problem AP (θ) . Theorem 1. Let Φ∗ρ be resolving for σ. Then σ yields the global minimum in the problem P (θ). The proof follows from Lemma 2. In conclusion, let us note that these optimality conditions are in the tradition of modification of Carathéodory and Krotov’s type conditions; we refer, for example, to [Clarke et al., 1998], [Krotov, 1996], [Milyutin & Osmolovskii, 1998], [Vinter, 2000], where optimal control problems with absolutely continuous tra- jectories were considered. Moreover, the optimality conditions stated by Theorem 1 are close to dynamic programming principle developed for impulsive processes in [Fraga & Pereira, 2008], [Motta & Rampazzo, 1996], [Pereira, Matos, & Silva, 2002], [Daryin & Kurzhanski, 2008]. 5 An Example Let us consider the optimal impulsive control problem J = V (t1 +) − y(t1 +) → min, (18) dy = (px1 + qx2 )dt, dx1 = a (1 − x1 ) µ1 , dx2 = c (1 − x2 /x1 ) µ2 , (19) y(0) = 0, x1 (0−) = x10 ∈ (0, 1), x2 (0−) = x20 ∈ (0, x10 ), V (θ−) ≤ R. (20) Here, parameters a, c, p, q are nonnegative, θ is a fixed point from (0, t1 ), µ = (µ1 , µ2 ) is a nonnegative ( )T Borel measure on [0, t1 ]. The vector fields generated by columns G1 = 0 a(1 − x1 ) 0 and G2 = ( )T ( ) 0 0 c(1 − x2 /x1 ) do not commutative. ( we need to use π(µ) = µ, γ(µ) , where γ(µ) = {ds , ωs (·)}s∈S , ) So, ∑ instead of only µ. As usual V (t) = |µc | [0, t] + ds , t ∈ (0, t1 ]. s≤t, s∈S Let us note that this problem may be interpreted as a model of the advertising expense optimization for two mutually complementary products in which an aggressive advertising campaign is possible. In [Dykhta & Samsonyuk, 2009] this problem was studied by using a maximum principle for impulsive processes. Let us consider one partial case of this problem. We assume that parameters satisfy the following conditions ( ) ( ) x∗1 < x∗∗ 1 , cq(t1 − θ) 1 − x20 /x∗∗ 1 < 1, cqt1 1 − x20 /x∗1 − β < 1, ( ) where x∗1 = 1 − (1 − x10 )e−aR , x∗∗ ∗ 1 = 1 − 1/ ap(t1 − θ) , β = pθa(1 − x1 ). Consider the impulsive process σ consisting of the control π(µ) with the components ( ( ) ) µ1 = Rδ(t) + (1/a) ln ap(t1 − θ)(1 − x10 ) − R δ(t − θ), µ2 = 0, ( ) ( ) S = {0; θ}, ds=0 = µ1 {0} , ds=θ = µ1 {θ} , ω s=0 (τ ) ≡ (1, 0), ω s=θ (τ ) ≡ (1, 0) and the corresponding trajectory x̄1 (0−) = x10 , x̄1 (t) = x∗1 , t ∈ (0, θ), x̄1 (t) = x∗∗ 1 , t ∈ [θ, t1 ], x̄2 (t) ≡ x20 . Then the optimality of σ is stated by using the strongly decreasing functions φ1 (t, x1 , x2 , V, y) = −(1 − x1 )eaV , t ∈ [0, θ), 1+β φ2 (t, x1 , x2 , V, y) = y − (1 + β)V − ln(1 − x1 ) − ptx1 + q(t1 − t)x2 , t ∈ [0, θ), a 1 φ3 (t, x1 , x2 , V, y) = y − V − ln(1 − x1 ) + p(θ − t)x1 + q(t1 − t)x2 , t ∈ [θ, t1 ]. a 502 One can prove that the set { φ1 , φ2 , φ3 } is resolving for σ. Acknowledgements This work was partially supported by the Comprehensive Program of Basic Research of SB RAS “Integration and Development”, Project 2017-II.2P/I.1-2. References [Bressan & Rampazzo, 1988] Bressan, A, & Rampazzo, F. (1988). 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