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.
This paper concerns an optimal impulsive control problem in which the control system is an extension of the control system
where
is the total variation on [a, t] for the function
w i (•). The symbol "a.e." signifies "almost everywhere with respect 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
Here, µ is a K-valued bounded Borel measure on T , x(•) is a function of bounded variation, B T is the set of all Borel subsets of T, and
, 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 solutions 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 [G i , G j ], 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) characterizes a way of approximation of µ by some sequences of L-absolutely continuous measures µ k = v k (t)dt, where v k : 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
Moreover, there exist constants c 1 , c 2 > 0 such that
Here, whose components satisfy the following conditions:
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
Given controls u(•), π(µ) and an initial condition x(a), consider the system of differential equations with the measure
) 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
Now we consider the impulsive control system (D)
Let u(•) and π(µ) satisfy ( 10). The tuple σ =
Here, functions
) satisfy ( 5)-( 8) with some initial condition x(a), comp(R n+1 ) is the set of non-empty compact subsets from R n+1 . Set by definition
Denote by Σ the set of all impulsive processes σ = ( X V , u(•), π(µ) ) of (D). Let us briefly comment on the relation between (D) and the corresponding conventional system (1), (2). Given X V , define its graph on T to be graph
Lemma 1 [Samsonyuk, 2015].
ii) for every k, the functions
Let θ = (θ 0 , . . . , θ k ) be a vector of fixed points of time such that a ≤ θ
∈ Σ, define the vector q σ that is composed of the one-sided limits of X V at the points θ j , j = 1, k; i.e.,
Let us consider the optimal impulsive control problem P (θ) with multipoint state constraints minimize
Here, C is a closed set in R d (qσ) , where d(q σ ) is the dimension of q σ , l : R d(qσ) → R is a continuous function. A process σ ∈ Σ is said to be a feasible process of P (θ) if q σ ∈ C.
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
Define the sets T , T Vα (t α , x α ), and Q φ (t α , x α , V α ) as follows
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 X V if for any (t 1 , x 1 , V 1 ) and (t 2 , x 2 , V 2 ), where (
Then φ is strongly decreasing if φ decreases along any X V ∈ 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 {t 0 , t 1 , . . . , t N } such that a = t 0 < t 1 < . . . < t N = b. Denote ∆ i = (t i−1 , t i ), 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
Here, H 0 (t, x, ψ) = max 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.
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.
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 X V ∈ T ∆i .
Next, for every t j , j = 0, N , we consider the so-called limiting system
with Lebesgue measurable controls ω(•). Let LT tj be the set of solutions of ( 14). Let us recall that a continuous function ξ(z, z V ) strongly decreases relative to (14) if ξ(z, z V ) is a solution of the proximal Hamilton-Jacobi inequality
(for more details see [Clarke et al., 1998]). Denote by Ξ tj the set of all continuous solutions of ( 15). Definition 2. The set
ρ is called a set of compound strongly decreasing functions.
In this section sufficient global optimality conditions for problem P (θ) will be formulated.
∈ Σ be a feasible process of P (θ) and let ρ = {t 0 , . . . , t N }, where a = t 0 < t 1 < . . . < t N = b, be some partition of T including all θ j , j = 0, k, i.e., ρ ⊇ {θ 0 , . . . , θ k }. Denote by I the set {j ∈ {0, . . . , N } | t j ∈ {θ 0 , . . . , θ r }} . In what follows we use the notation
Given ρ, define the sets
Let us note that the set R ρ 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 q σ,ρ belongs to R ρ . And the contrary, for any q ∈ R ρ there exists σ ∈ Σ such that q σ,ρ = q.
} be an arbitrary set of compound strongly decreasing functions. Define the sets
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 Z tj , j = 0, N , respectively; i.e.,
) 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
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 trajectories 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].
Let us consider the optimal impulsive control problem
Here, parameters a, c, p, q are nonnegative, θ is a fixed point from (0, t 1 ), µ = (µ 1 , µ 2 ) is a nonnegative Borel measure on [0, t 1 ]. The vector fields generated by columns
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
. Consider the impulsive process σ consisting of the control π(µ) with the components
, ω s=0 (τ ) ≡ (1, 0), ω s=θ (τ ) ≡ (1, 0)
and the corresponding trajectory
Then the optimality of σ is stated by using the strongly decreasing functions φ 1 (t, x 1 , x 2 , V, y) = −(1 − x 1 )e aV , t ∈ [0, θ),
One can prove that the set { φ 1 , φ 2 , φ 3 } is resolving for σ.