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 x˙ (t) = f (t, x(t), V (t), u(t))+ G(t, x(t), V (t))v(t), u(t) ∈ U, v(t) ∈ K
a.e. on T, (1) (2) (3) 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 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 [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) characterizes 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
0 ds ≥ ||µ({s})||, ωs(τ )dτ = µ({s}), (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 dzV s(τ ) dτ = 1,
∑ = G(s, zs(τ ), zV s(τ ))ωs(τ ), dV (t) = |µc| + (zV s(ds) − V (s−))δ(t − s),
V (a) = 0, zs(0) = x(s−), zV s(0) = V (s−), τ ∈ [0, ds], s ∈ S.
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 Now we consider the impulsive control system (D)
V (t) = |µc|([a, t])+
ds, t ∈ (a, b].
∑ s≤t, s∈S dx(t) = f (t, x(t), V (t), u(t))dt + G(t, x(t), V (t))π(µ),
u(t) ∈ U a.e. on T, π(µ) ∈ W(T, K).
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 ii) for every s ∈ S
XV (t) = {(x(t), V (t))},
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 (t+) = {(x(t+), V (t+))} ∀ t ∈ [a, b),
XV (a−) = {(x(a), 0)}, 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(Rn+1). Denote by d(A, B) the Hausdorff distance between A and B.
Lemma 1 [Samsonyuk, 2015]. 1) Let σ = (XV , 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))
T T → 0, k → ∞. 2) Let {xk(·), Vk(·), uk(·), vk(·)} be a sequence of functions such that i) sup ||vk(·)||L1 < ∞;
k iii) {xk(a)} is bounded. ii) for every k, the functions xk(·), Vk(·), uk(·), vk(·) satisfy (1), (2); Then, there exist σ = (XV , u(·), π(µ)) ∈ Σ and a subsequence {xkj(·), Vkj(·), ukj(·), vkj(·)} such that d( graph XV , graph (xkj, Vkj))
T T → 0,
j → ∞. (7) (8) (9) (10) (11)
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 subject to
J (σ) = l(qσ) σ ∈ Σ, qσ ∈ C.
Here, C is a closed set in Rd(q ), where d(qσ) is the dimension of qσ, l : Rd(q ) → R is a continuous function. A process σ ∈ Σ is said to be a feasible process of P (θ) if qσ ∈ C. 3
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 , TV (tα, xα), and Qφ(tα, xα, Vα) as follows T = {XV | ∃ σ = (XV , u(·), π(µ)) ∈ Σ},
TV (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)}.
De nition 1. Function φ is strongly decreasing relative to (D) if for any (tα, xα) ∈ T × Rn, Vα ≥ 0 and for any XV ∈ TV (tα, xα) the inclusion graph XV ⊂ Qφ(tα, xα, Vα) [t ,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 pV + H1(t, x, px) ≤ 0 ∀ p = (pt, px, pV ) ∈ ∂P φ(t, x, V ), ∀ (t, x, V ) ∈ (ti−1, ti) × Rn × [0, +∞), ∀ p = (pt, px, pV ) ∈ ∂P φ(t, x, V ), ∀ (t, x, V ) ∈ [ti−1, ti] × Rn × (0, +∞). (12) (13) Here, H0(t, x, ψ) = max⟨ψ, f (t, x, u)⟩, H1(t, x, ψ) = ωm∈aKx1⟨ψ, G(t, x)ω⟩, the set ∂P φ(t, x, V ) is the proximal u∈U 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. 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
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, +∞)
De nition 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 , {Ξt∗j }j=0,N .
Then Φ∗ρ is called a set of compound strongly decreasing functions. 4
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 qσj0 =. XV (tj−), qσj1 =. XV (tj+), j = 0, N , . ({qσj0, qσj1}j=0,N .
) qσ,ρ =
X∆i = {((xi−1, Vi−1), (xi, Vi))
∃ XV ∈ T : XV (ti−1+) = (xi−1, Vi−1), XV (ti−) = (xi, Vi) }, i = 1, N , Ztj = { ((z0, zV 0), (z1, zV 1)) ∃ (z(·), zV (·)) ∈ LTtj :
. z(0) = z0, z(d) = z1, zV (d) − zV (0) = d, d = zV 1 − zV 0 } (qi−1, q0i) ∈ X∆i , i = 1, N , (q0j, q1j) ∈ ZEtj , j = 0, N 1 } .
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 ∈ Rρ there exists σ ∈ Σ such that qσ,ρ = q.
Let Φ∗ρ = {{Φ∗∆i }i=1,N , {Ξt∗j }j=0,N } be an arbitrary set of compound strongly decreasing functions. Define the sets
∩ φ∈Φ∆i ξ∈Ξtj A[Φ∗∆i ] =
{(q1, q0)| q0, q1 ∈ Rn × R+, φ(ti, q0) − φ(ti−1, q1) ≤ 0}, LA[Ξt∗j ] =
∩ {(q0, q1) | q0, q1 ∈ Rn × R+, ξ(q1) − ξ(q0) ≤ 0}, A[Φ∗ρ] = {q = ({q0j, q1j}j=0,N ) (q1i−1, q0i) ∈ A[Φ∗∆i ], i = 1, N , (q0j, q1j) ∈ LA[Ξt∗j ], j = 0, N } .
By using the strong decreasing property of functions from Φ∗∆i , i = 1, N , and Ξt∗j , j = 0, N , one can readily obtain that A[Φ∗∆i ], i = 1, N , and LA[Ξt∗j ], 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[Ξt∗j ], j = 0, N .
(17) (14) (15) (16) j = 0, N , i = 1, N , j = 0, N ,
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 = (XV , u, π(µ)) be an examining process of P (θ) and qσ be the corresponding vector of the one-sided limits of XV . Let the vector qσ,ρ be defined by (16).
The set Φ∗ρ is said to be resolving for σ if the vector qσ,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 Carath´eodory 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]. 5
Let us consider the optimal impulsive control problem
J = V (t1+) − y(t1+) → min, dy = (px1 + qx2)dt, dx1 = a (1 − x1) µ1,
dx2 = c (1 − x2/x1) µ2, y(0) = 0, x1(0−) = x10 ∈ (0, 1), x2(0−) = x20 ∈ (0, x10),
V (θ−) ≤ R.
(18) (19) (20) Here, parameters a, c, p, q are nonnegative, θ is a fixed point from (0, t1), µ = (µ1, µ2) is a nonnegative Borel measure on [0, t1]. The vector fields generated by columns G1 = ( 0 a(1 − x1) 0 )T and G2 = ( 0 0 c(1 − x2/x1) )T do not commutative. So, we need to use π(µ) = (µ, γ(µ)), where γ(µ) = {ds, ωs(·)}s∈S , 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 x1∗ < x1∗∗, cq(t1 − θ)(1 − x20/x1∗∗) < 1, cqt1(1 − x20/x1∗)− β < 1, where x1∗ = 1 − (1 − x10)e−aR, x∗∗ = 1 − 1/(ap(t1 − θ)), β = pθa(1 − x1∗). Consider the impulsive process σ 1 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}), ωs=0(τ ) ≡ (1, 0), ωs=θ(τ ) ≡ (1, 0) and the corresponding trajectory x¯1(0−) = x10, x¯1(t) = x∗, t ∈ (0, θ), 1 x¯1(t) = x1∗∗, t ∈ [θ, t1], x¯2(t) ≡ x20.
Then the optimality of σ is stated by using the strongly decreasing functions φ1(t, x1, x2, V, y) = −(1 − x1)eaV , 1 + β
a φ2(t, x1, x2, V, y) = y − (1 + β)V −
ln(1 − x1) − ptx1 + q(t1 − t)x2, 1 φ3(t, x1, x2, V, y) = y − V − a ln(1 − x1) + p(θ − t)x1 + q(t1 − t)x2, t ∈ [0, θ), t ∈ [0, θ), t ∈ [θ, t1].
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.