Optimal control problems for a neutral integro-differential system with infinite delay

This work devotes to the study on problems of optimal control and time optimal control for a neutral integro-differential evolution system with infinite delay. The main technique is the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature. We first establish the existence and uniqueness of mild solutions and discuss the compactness of the solution operator for the considered control system. Then, we investigate the existence of optimal controls for the both cases of bounded and unbounded admissible control sets under some assumptions. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. An example is given at last to illustrate the applications of the obtained results.

1. Introduction. In this paper, we consider the following neutral integrodifferential evolution systems with infinite delay: where the state function x(·) ∈ X and the control function u(·) ∈ U with X and U being two Hilbert spaces. The history functionals x t (·) : (−∞, 0] → X given, in the usual way, by x t (θ) = x(t + θ) for θ ≤ 0, belong to some abstract phase space B defined axiomatically. The (unbounded) linear operator A generates an analytic semigroup on X, γ(t) is a family of closed linear operators on X with domain D(γ(t)) ⊇ D(A) and N (t) in the neutral term is a family of bounded linear operators on X for every t ≥ 0. Additionally, {B(t) : t ≥ 0} is a family of bounded linear operators from U to X and F : [0, T ] × B → X is a nonlinear Lipschitz continuous function to be described below.
As well known, the concept of optimal control is one of the fundamental concepts in mathematical control theory for infinite differential systems [24,25]. Roughly speaking, optimal control generally describes that the minimization for a cost function of the states and control inputs of the system over a set of admissible control functions. Meanwhile, the time optimal control problem is to find a control which transfers the trajectory of a control system from a given initial state to a specified final state in minimum time. Actually, the optimal and time optimal control have been commonly used in control theory. Hence, the problems of optimal and time optimal control for infinite-dimensional evolution systems gain much more attention in the past decades. In [13], Harrat et al. established some sufficient conditions for solvability and optimal controls of an impulsive nonlinear Hilfer fractional delay evolution inclusion in Banach spaces. Mokkedem and Fu [28] discussed the standard optimal control and time optimal control problems for a class of semilinear evolution systems with infinite delay by employing the theory of fundamental solution as in papers [18,19,30]. While Tucsnak et al. studied in [36], by weakening the regularity assumptions on the initial data with z 0 ∈ X, the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator. For more related works on optimal and time optimal control problems for evolution equations, we refer to [1,2,3,20,22,23,26,29,33,38,39,40] and the references cited therein.
On the other hand, System (1) is an abstract form of neutral partial functional integro-differential equations (NPFIDEs) with infinite delay. Indeed, NPFIDEs can be used to describe a lot of natural phenomena arising from many fields such as electronics, fluid dynamics, chemical kinetics and biological models. An effective way of studying NPFIDEs is to transfer them into integro-differential evolution equations with or without delay in abstract spaces. In [9,10,11], Grimmer et al. proved the existence of solutions of the following integro-differential evolution equation in Banach space X: where f : R + → X is a continuous function. The author(s) showed the representation, existence and uniqueness, of solutions of (2) via the resolvent operator associated to the following linear homogeneous equation That is, the resolvent operator, replacing the role of C 0 -semigroup for evolution equations, plays an important role in solving Eq. (2) in weak and strict senses. By means of the theory of resolvent operators, in these years much work on various topics, such as existence results, asymptotic properties, controllability and optimal control on various semilinear integro-differential systems have been investigated by many authors, see [4,5,15,34,39,40], for example. In particular, Diallo et al. [4] obtained by using the theory of resolvent operator and set-valued mapping the existence and stability for the optimal control problem of the following semi-linear integro-differential equation with compact control set In [39] Yan and Lu also discussed the optimal control problem of semi-linear fractional stochastic integro-differential equation with infinite delay. The main tool there was the theory of analytic α−resolvent operators. Very recently, Dos Santos et al. [7,14] established the theory of resolvent operators for neutral linear integro-differential equations (to be presented briefly in Section 2). It is clear that this theory of resolvent operators is a routine extension of that introduced by Grimmer et al. [9,10,11] and has a more comprehensive description for various dynamic behaviors of neutral integro-differential equations. For example, Dos Santos applied it in [6] to study the existence of solutions for abstract neutral integro-differential equations with state-dependent delay. Likewise, by using the theory of new resolvent operators and approximating technique, Jeet and Sukavanam [17] obtained the approximate controllability for nonlocal and impulsive neutral integro-differential equations. Up to now, there are many papers on neutral integro-differential equations by employing this new theory, see [32,37].
In this article we intend to derive the existence of optimal and time optimal controls of the infinite-delayed neutral integro-differential system (1). Our purpose here is to develop the results for semi-linear differential equations established in [28] to neutral integro-differential equations. Certainly, neutral integro-differential equations are much more complicate than semi-linear differential systems to be dealt with. The essential technique in all our discussion, other than in [39], is the theory of resolvent operators for linear neutral integro-differential systems established in [7,14]. As the considered system involves a nonlinear function, we will first prove the compactness of the solution operator W mapping from each control u(·) to its corresponding mild solution. Then we obtain the existence of optimal and time optimal controls for the control system (1) by using the compactness of W and standard convergence arguments. Due to the compactness of W , as in [28], we do not require the convexity assumptions on the integral cost function. Note that the convexity assumptions on integral cost functions were generally adopted in literature, see [2,13,20,22,30,38,40] for example. It is also worth stressing here that, to the best of our knowledge, the study of the optimal control problem for abstract neutral integro-differential with infinite delay of the form (1) is an untreated topic in the literature, this is actually an additional motivation of our work. Clearly our work can be regarded as extension and development of that in [4,28,30,38] and other related papers mentioned above.
The organization of this work is as follows. Some basic notations and preliminary facts about the resolvent operators and phase space for B for infinite-delayed equations are presented in Section 2. In Section 3, using the Banach contraction principle, we first prove the existence and uniqueness of mild solutions for System (1) represented via the resolvent operator. Then, we show the compactness of the solution operator W . We discuss in Section 4 the existence of optimal controls of certain fixed time integral cost function subject to the neutral integro-differential control system (1) in two cases of bounded and unbounded admissible control sets, respectively. In Section 5, we study the existence of optimal control which transfer the mild solutions of the neutral integro-differential control system (1) from the initial data to a target set in the shortest time, namely, the time optimal control to a target set. Following the existence results, we give a theorem about the convergence of time optimal controls to a point target set. Finally, an example is provided to show the applications of the obtained results in Section 6.
2. Preliminaries. Let X and K be two separable Hilbert spaces and L (X; K) stands for the space of all bounded linear operators from X into K, we abbreviate it to L (X) whenever K = X. For the closed linear operator (A, D(A)) on X in Eq. (1), denote by Y the Banach space (D(A), · ) with the graph norm · Y given by We next state briefly the theory of resolvent operators for linear neutral integrodifferential equations which was introduced in [7,14].
Definition 2.1. (see [7]) A one parameter family of bounded linear operators (R(t)) t≥0 on X is called a resolvent operator for if the following conditions are verified.
In the sequel we always impose the following hypotheses on the operators appearing in the system (1) or (3).
for each x ∈ D and all λ ∈ Λ θ . Then, it follows from [7,14] that, under these conditions, there is a family of resolvent operator R(t) t≥0 for linear neutral integro-differential system (3) (the linear part of System (1)) defined by for λ ∈ Λ r,ϑ and some N > 0. Here r > 0, ϑ ∈ ( π 2 , θ) are fixed numbers, and the contour Γ r,ϑ can selected to be included in the region Λ r, r,ϑ and Γ 2 r,ϑ . The following theorem summarizes several important properties of the resolvent operator R(t).
Proof. Assertions (i) and (ii) were established in [7], while Assertions (iii) and (iv) come from Lemma 2.6 and Lemma 4.1 in [17], respectively. Now we turn to introduce the axiomatic definition of the phase space B introduced by Hale and Kato [12], adopting the terminologies used in Hino et al. [16]. That is, B is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm · B , which satisfies the following axioms: for every t ∈ [σ, σ + a] the followings hold: For example, let the phase space B = C r × L p (g : X), r ≥ 0, 1 ≤ p < ∞, which consists of all classes of functions ϕ ∈ (−∞, 0] → X such that ϕ is continuous on [−r, 0] and g ϕ(·) p is Lebesgue integrable on (−∞, −r), where g : (−∞, −r) → R is a positive Lebesgue integrable function. The seminorm in B is defined by Then it satisfies all the axioms (A), (A 1 ) and (B), see [16].
3. Existence and uniqueness of mild solutions. In this section, we prove the existence and uniqueness of mild solutions for System (1) and discuss the compactness of the solution operator. In what follows, the control function u ∈ L p (0, T ; U ) with p > 1, and ϕ ∈ B will be a given initial function. We introduce the functions and we always assume that Then the mild solutions of System (1) are defined as To carry on our discussion, we make the following assumptions on Eq. (1). (H 1 ) The operator B(·) ∈ L ∞ (0, T ; L (U, X)) and B(·) ∞ stands for its usual norm on L ∞ (0, T ; L (U, X)).
is Lipschitz continuous with respect to the second variable, i. e., there exist a positive constant L > 0 such that and there holds Now we establish the result on existence and uniqueness of mild solutions of the control system (1) as follows. First, for the functions K(·) and M (·) in Axiom (A)(iii) we put, for any b ∈ (0, T ], Take some T 1 ∈ (0, T ] such that Denote and defined the set E(T 1 , ρ 1 ) by It is clear that E(T 1 , ρ 1 ) is a closed, bounded and convex subset of C([0, T 1 ]; X).
To check the relative compactness of {x(t; u n ), n ≥ 1} in the space X for each t ∈ [0, T ], it suffices to show it on (0, T ] since {x(0; u n ), n ≥ 1} = {ϕ(0)}. Let t ∈ (0, T ] be fixed. Then, for any u n ∈ L p (0, T ; U ), we get that Let > 0 be sufficiently small such that t − > 0 and define Observe that < +∞, from which and the fact R( ) is compact, we infer that, for each t ∈ (0, T ], the set is relatively compact in X and hence the set {x (t; u n ), n ≥ 1} does so. By Theorem 2.2 (iv) we deduce easily that →0 (as → 0), it follows immediately that the set {x(t; u n ), n ≥ 1} is relatively compact in X as well for every t ∈ (0, T ]. Accordingly, by Ascoli-Arzela theorem, we conclude that the solution operator W is compact from L p (0, T ; U ) to C([0, T ]; X). This completes the proof. Now, we define the so-called Nemitsky operator F : L p (0, T ; U ) → L p (0, T ; X) corresponding to the nonlinear function F (·, ·) by the formula Then using the similar arguments as the proof of Lemma 3.2 in [28], we can establish the following lemma. 4. Optimal control. The purpose of this section is to discuss the optimal control problem for System (1). Let the admissible set U ad be a closed convex subset of L p (0, T ; U ). As above, we denote the mild solutions of Eq. (1) by x(·; u) to express their dependence on u ∈ U ad . Now, define the integral cost function as J = J (u, x(·; u)) := φ 0 (x(T ; u)) + T 0 [φ 1 (t, x(t; u), x t (θ; u)) + φ 2 (t, u(t))] dt, (11) where the kernel functions φ i , i = 0, 1, 2, satisfy that for each u ∈ U ad and the functional Γ : U ad → R given by is continuous and convex. We consider here the problem of optimal control which is described as follows. (P 1 ) Find a control u ∈ U ad which minimizes the cost function J subject to the constraint (1). Such a control u ∈ U ad is called an optimal control and the pair (u, x(·; u)) is called an optimal solution for J .
Subsequently, we study the problem (P 1 ) in two cases, one is the case that the admissible set U ad is bounded and the other case is that U ad is unbounded in L p (0, T ; U ). We first have the following existence result of (P 1 ) for the case of bounded admissible set U ad .
are all satisfied and U ad is bounded in L p (0, T ; U ). If the resolvent operator (R(t)) t≥0 is compact, then the problem (P 1 ) admits at least one solution, i. e., there exists at least one control u ∈ U ad which minimizes the cost function J subject to (1).
From the definition of infimum there exists a minimizing sequence of feasible pair {(u n , x(·; u n ))} n≥1 ⊂ A ad := {(u, x(·; u)) | u ∈ U ad }, such that J (u n , x(·; u n )) → m as n → +∞. Since {u n } n≥1 ⊆ U ad is bounded in L p (0, T ; U ), there exists a subsequence, relabeled still as {u n } n≥1 , and u 0 ∈ L p (0, T ; U ) such that u n → u 0 weakly in L p (0, T ; U ). (12) Noting that U ad is closed and convex, by Marzur lemma we have u 0 ∈ U ad as well. Rewrite x(·; u n ) as x n (·) := (n ≥ 1) (the mild solutions of System (1) corresponding to u n ), then x n (·) satisfy the following integral equation Denote by x 0 the mild solution of Eq. (1) corresponding to the control u 0 , i. e., Then from (12) and the compactness of the operator W (Theorem 3.3) it follows that x n (t) → x 0 (t) strongly in X, n → +∞, (13) by taken a subsequence of {x n (·)} n≥1 if necessary.
Meanwhile, combining (A)(iii) and (H 3 )(ii) yields readily that from which and Fatou's lemma we further derive that lim inf In addition, since the assumption (H 3 )(iii) implies that the functional Γ is weak lower semi-continuous, namely, It hence follows from (14), (15) and (16) that which means that J attains its infimum at u 0 , x 0 ∈ A ad . The proof is proved.

Remark 1.
We would like to emphasize that, since we have shown the compactness of the operator W , we do not require the functions φ 0 and φ 1 in the cost function J satisfy the convex condition which was used commonly in the literature such as [2,13,20,22,30,38,40].
Now we consider the case that U ad is unbounded in L p (0, T ; U ). To this end, besides the previous assumptions we also suppose that the followings hold true. (H 4 ) (i) There exists a constant c 0 > 0 such that φ 0 (·) ≥ −c 0 on X.
(iii) There exists a monotonely increasing function θ 0 ∈ C(R + ; R) such that lim r→∞ θ 0 (r) = +∞ and Under these conditions, we can prove the following result for the case that the admissible set U ad is unbounded.
Theorem 4.2. Assume that the conditions (H 1 ) − (H 4 ) are all verified and U ad is unbounded in L p (0, T ; U ). If the resolvent operator (R(t)) t≥0 is compact, then there exists at least one optimal control u ∈ U ad that minimizes the cost function J subject to (1).
Note that, by virtue of (H 4 ), and lim r→∞ θ 0 (r) = +∞, we see that the minimizing sequence {u n } n≥1 is bounded in L p (0, T ; U ). Hence, conducted the similar arguments as in the proof of Theorem 4.1, the assertion follows.

5.
Time optimal control. In this section, we study the time optimal control problem for System (1). Let the admissible set U ad and the set W be bounded, closed and convex respectively in L p (0, T ; U ) and X. We set and suppose that U 0 = ∅. For each u ∈ U 0 , we define the transition time to be the first timet(u) such that x(t; u) ∈ W and the set W is called a target set. Then the time optimal control problem considered in this part is described as (P 2 ) Find a control u ∈ U 0 such that subject to the constraint (1). In (P 2 ), such a u ∈ U 0 is called a time optimal control andt(u) is called an optimal time. Now we solve the problem (P 2 ), namely, we prove the existence of a control which transfers the mild solutions of the constraint (1) from the initial data to a target set in the shortest time.
Theorem 5.1. Let ϕ ∈ B. Suppose that (H 1 ) and (H 2 ) hold and U 0 = ∅. If the resolvent operator (R(t)) t≥0 is compact, then there exists a time optimal control u ∈ U 0 for the problem (P 2 ).
(1) corresponding to the controls u n . Assume that t n :=t(u n ) ↓ t 0 (n → +∞). Then, x n (t n ) satisfies the integral equation Clearly, since W and U ad are bounded, closed and convex subsets in the Hilbert spaces X and L p (0, T ; U ) respectively, {x n (t n )} n≥1 is a bounded sequence in W , and moreover, there existx ∈ W,ũ ∈ U ad and subsequences of {x n (t n )} n≥1 and {u n } n≥1 , still denoted by themselves, such that u n →ũ weakly in L p (0, T ; U ).
We subsequently show thatũ is the time optimal control for (P 2 ) with the optimal time t 0 . In fact, it is clear that I 1 = R(t n )ϕ(0) → R(t 0 )ϕ(0) as n → +∞, and meanwhile, from (19) and Lemma 3.4 we have On the other hand, we find that →0 as n → +∞.
Hence, taking the limit for n → +∞ on both sides of (17), it then yields in X. Combining this with (18) we derive straightly that which manifests thatũ ∈ U 0 . It is clear by definition of optimal time that t 0 = t(ũ) ≤t(u) for all u ∈ U 0 . Consequently,ũ is the time optimal control for (P 2 ), which is our desired result.
Next, as in [30], we also consider the case in which the target set W is singleton. Put W = {w 0 } such that ϕ(0) = w 0 . Since X is reflexive, we can choose a decreasing sequence of non-empty, bounded, closed and convex sets {W n } n≥1 in X such that w 0 = +∞ n=1 W n and dist(w 0 , W n ) = sup x∈Wn |x − w 0 | → 0 as n → +∞. (20) Assume that, for each n ≥ 1, Then the time optimal control problem (P 2 ) with the target set {w 0 } can be solved as follows.
Theorem 5.2. Let {W n } n≥1 be a decreasing sequence of non-empty, bounded, closed and convex sets in X satisfying (20) and (21). Let {u n } n≥1 be a sequence such that each u n is the time optimal control with the optimal time t n to the target set W n . Then there exists a time optimal control u 0 with the optimal time t 0 = sup x n (t n , u n ) ∈ W n → w 0 strongly in X.
Thus, using similar arguments as in the proof of Theorem 5.1, we can easily prove that Moreover, we can verify that u 0 is the time optimal control and t 0 is the optimal time to the target set {w 0 }. Indeed, if it is not true, then there exists a control v ∈ U ad such that x t 1 ; v = w 0 with t 1 < t 0 . We may choose an integer n 0 such that t 1 < t n0 ≤ t 0 . From the definition of U n0 0 there must have v ∈ U n0 0 . On the other hand, u n0 is the time optimal control with the optimal time t n0 to the target set W n0 . Hence t n0 ≤t(u) for all u ∈ U n0 0 . Particularly, t n0 ≤t(v) ≤ t 1 , which is a contradiction and hence the desired result follows.
6. An example. In this section, we apply the obtained results to investigate the optimal control problems for the following neutral partial integro-differential control system with infinite delay.
where a(·), b(·), c(·, ·), f (·, ·), r(·, ·, ·), u(·, ·) and ϕ(·, ·) are functions to be described below. System (22) arises in the study of heat flow in materials of the so-called fading memory type ( [27]). Here, z(t, x) represents the temperature of the point x at time t. Various topics on this system have been studied in literature in the past decades. For instance, in [21] and [35] the authors have discussed respectively the existence and maximal regularity of solutions of this kind of equations. However, little is known on its optimal control problems. Here we can use the above results to obtain the optimal control and time optimal control for System (22) under some proper conditions. To represent this problem as the form of Eq. (1), we take X = U = L 2 ([0, π]) and we define Z(t)(·) := z(t, ·) and ϕ(t)(·) := ϕ(t, ·). Let A : D(A) → X be the operator given by Then A generates a strongly continuous semigroup (S(t)) t≥0 which is analytic, compact and self-adjoint. Furthermore, A has a discrete spectrum, the eigenvalues are −n 2 , n ∈ N + , with the corresponding normalized eigenvectors e n (x) = 2 π sin(nx), n = 1, 2, · · · . Then the following properties hold: −n 2 ξ, e n e n .
(ii) For every ξ ∈ X, e −n 2 t ξ, e n e n .
Here we take the phase space B = C 0 ×L 2 (g : X) which was described in Section 2 with r = 0 and p = 2. Thus the norm of B is given by, for ϕ ∈ B, It is known that, by choosing a proper function g, the phase space C 0 × L 2 (g : X) satisfies the Axioms (A), (A 1 ) and (B).
Then under these notations System (22) is rewritten into the form Subsequently let us certify that for this system the conditions in Theorem 3.2 are all fulfilled. Firstly, the assumptions (a 3 ) − (a 5 ) ensure that the function F satisfies the hypotheses (H 2 ) with L = l √ k. Actually, from the assumptions (a 3 ) − (a 5 ) we compute that, for any t ∈ [0, T ] and ϕ 1 , ϕ 2 ∈ B, |c(t, t + θ)| |f (t + θ, ϕ 1 (θ, x)) − f (t + θ, ϕ 2 (θ, x))| dθ Then from Theorem 5.1 there exists a control u ∈ U 0 such that t(u) ≤t(u), for all u ∈ U 0 , subject to the constraint (22), i. e., there exists a time optimal control to the target set W subject to System (22). Particularly, if W = {w 0 } (⊂ X) is a singleton so that (21) is verified, then, by Theorem 5.2, there also exists a time optimal control to the point target set {w 0 } subject to System (22).