TIME-VARYING INTEGRO-DIFFERENTIAL INCLUSIONS WITH CLARKE SUB-DIFFERENTIAL AND NON-LOCAL INITIAL CONDITIONS: EXISTENCE AND APPROXIMATE CONTROLLABILITY

. In this paper, we mainly consider a time-varying semi-linear integrodiﬀerential inclusion with Clarke sub-diﬀerential and a non-local initial con- dition. By a suitable Green function combined with a resolvent operator, we ﬁrstly formulate its mild solutions and show that it admits at least one mild so- lution which can exist in a well-deﬁned ball with a radius big enough. Through constructing a proper functional, we then derive a useful characterization of the approximate controllability for its related linear system in Green function terms, and establish a suﬃcient condition for the approximate controllability of the time-varying semi-linear integro-diﬀerential inclusion. Lastly, we also consider the ﬁnite approximate controllability of the time-varying semi-linear integro-diﬀerential inclusion via variational method.

1. Introduction. In this paper, we consider the following time-varying semi-linear integro-differential inclusion with Clarke sub-differential and a non-local initial condition a(t, s)u(s)ds + Bv(t) + ∂G(t, u(t)), t ∈ I, where I := [0, b], the operator A (t) with its domain D independent of t, and a(t, s) ∈ L(H), 0 ≤ s ≤ t ≤ b, generate a integral resolvent operator R(t, s) in a Hilbert space H (see Definition 2.2), B is a bounded linear operator from a Hilbert space V to H, the notation ∂G(t, ·) denotes the Clarke sub-differential of G(t, ·), the state u(·) takes its value in H, the control v(·) is given in L 2 (I, V), and 0 < t 1 < t 2 < · · · < t m < b, m is a positive integer, c k = 0, k = 1, 2, · · · , m are real numbers. Partial integro-differential equations are often used to deal with models of viscoelastic materials or materials with memory ( [8,32]). Under certain conditions, those partial integro-differential equations can be treated as abstract evolution equations by resolvent operators without semi-group properties ( [17,18]). It is remarkable that the resolvent operator with fixed point method is an effective approach for the solvability of abstract integro-differential equations. For example, RaviKumar [37] proved the existence of semi-classical and mild solutions of nonlinear integro-differential equations with nonlocal conditions in Banach spaces via analytic resolvent operators and Schaefer's fixed point theorem. Ezzinbi and Ghnimi [15] considered existence and regularity of solutions for a neutral partial functional integro-differential equation. Dieye, Diop and Ezzinbi [13] studied the existence and exponentially stability in p-mean square for some stochastic integrodifferential equation with delays. Lizama and N'Guérékata [24] presented a general operator theoretical approach to study bounded mild solutions for a semi-linear integro-differential equation in Banach spaces. Chang and Ponce [6] investigated uniform exponential stability and the existence and uniqueness of bounded mild solutions to a semi-linear integro-differential equation. Sometimes, when treated some partial integro-differential equations in parabolic cases, they can usually be formulated as time-varying (or non-autonomous) abstract evolution equations. See for instance, RaviKumar [38] dealt with regularity of solutions of evolution integrodifferential equations with deviating argument. Liu and Ezzinbi [21] obtained the existence and uniqueness of mild and classical solutions to a non-autonomous integro-differential equation with non-local initial conditions. Non-local initial conditions originate from physical phenomena. It is shown that, the non-local initial condition u(0) = m k=1 c k u(t k ) can be more suitable than the classical initial condition u(0) = u 0 in some physical models. For instance, Deng [12] applied the non-local condition u(0) = m k=1 c k u(t k ) to describe the diffusion phenomenon of a small amount of gas in a transparent tube. Under such circumstance, this nonlocal condition is involved in some additional measurements at t k , k = 1, 2, · · · , m, which is more precise than the measurement just at t = 0. It was also suggested by Byszewski in [5] that, if c k = 0, k = 1, 2, · · · , m, then the results could be used to kinematics to determine the location evolution t → u(t) of a physical object for which the specifical positions u(0), u(t 1 ), u(t 2 ), · · · , u(t m ) may be unknown, but the non-local condition u(0) = m k=1 c k u(t k ) does hold. The importance of non-local initial conditions to other types of equations can be found in [25,40,4,44,19] and references therein.
The concept of controllability plays a crucial role in design and analysis of control systems. Exact controllability and approximate controllability are two important concepts in mathematical control theory [3]. Exact controllability means that the addressed system can be steered to arbitrary final state whereas approximate controllability enables us to steer the system to arbitrarily small neighborhood of final state. In general, it is difficult to realize exact controllability of control systems in infinite dimensional spaces or some special systems in finite dimensional spaces (see [1,2,26]). From a practical point of view, the approximate controllability becomes a more natural concept. Therefore, there are interesting works on approximate controllability of different systems modelled by evolution equations, integrodifferential equations, functional differential equations, differential inclusions and fractional evolution equations ( [45]) in infinite dimensional spaces. See for instance, Abbas et al [31] established approximate controllability of sub-diffusion equation with impulsive condition. Sakthivel et al [39] considered approximate controllability of fractional stochastic differential inclusions with non-local conditions. Yan and Jia [43] studied approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay. Fu [16] discussed the approximate controllability of semi-linear non-autonomous evolution systems with state-dependent delay. Xiao and Zhu [41] considered approximate controllability for a second order semi-linear impulsive functional differential inclusion in Hilbert spaces. Finite approximate controllability is a stronger property than that of approximate controllability. Lions and Zuazua [20] have proved that finite approximate controllability is a consequence of approximate controllability in the context of linear heat equation while one property may not be deduced as a consequence of the other one in the nonlinear context. Recently, Mahmudov [28] has developed the notion of finite approximate controllability to a semi-linear evolution equations in Hilbert spaces, which implies that the control can be selected such that the final state is not only approximately controllable but also satisfies simultaneously a finite number of exact constraints. The concept of finite approximate controllability has further been extended to some semi-linear fractional evolution systems in Hilbert spaces [29,30].
The time-varying integro-differential inclusion problem (1) has a closed relation to the following H-variational inequality problem for ∀µ ∈ H, t ∈ I, where G 0 (t, ·; ·) represents the Clarke directional derivative of G(t, ·) (see Sect. 2). In fact, if there exists a function g ∈ L 2 (I, H) such that g(t) ∈ ∂G(t, u(t)) and a(t, s)u(s)ds + Bv(t) + g(t), t ∈ I, then for ∀µ ∈ H and a.e. t ∈ I, we have Noticing that g(t) ∈ ∂G(t, u(t)) and g(t), µ H ≤ G 0 (t, u(t); µ) (see also Sect. 2), we can see that for ∀µ ∈ H, the H-variational inequality (2) holds true. So, we can study the H-variational inequality (2) through investigating the problem (1). Hvariational inequality initiated by Panagiotopoulos, has been regarded as one of the most powerful tools to deal with non-smooth and non-convex energy superpotentials problems in mechnics [35,36]. Solvability, well-posedness and optimal controls of some H-variational inequalities are also investigated under different conditions in mathematics, we can refer to Lu and Liu [27], Huang and Xiao [42], Motreanu and Motreanu [34], Migórski and Sofonea [33], Chang and Pei [7], and references cited therein for more details. Approximate controllability of control systems represented by H-variational inequalities was first studied in [22] by Liu and Li, in which sufficient conditions were established for existence and approximate controllability of a semi-linear time-invariant system with classical initial conditions. Particularly, let a(t, s) ≡ 0 with a classical initial condition u(0) = u 0 in (2), it is reduced to a semilinear evolution H-variational inequality problem, the approximate controllability of which was investigated in [23] via a two-parameter evolution system having a semi-group property. However, there is still little information on approximate controllability of time-varying system with a non-local initial condition described by the integro-differential H-variational inequality (2). Consequently, it is of interest to consider the problem (1). Inspired by above mentioned works, the main purpose of this paper is to investigate approximate controllability of the time-varying system (1). By introducing a well-defined Green function combined with a resolvent operator, we firstly give the expression of its mild solutions and prove that the problem (1) admits at least one mild solution which can exist in a suitable ball with a radius big enough. Through constructing a proper functional, we derive a useful characterization of the approximate controllability for its related linear system in Green function terms, and establish a sufficient condition for the approximate controllability of (1). Finally, we also investigate the finite approximate controllability of (2) via variational method.
The structure of this paper is as follows: Sect. 2 is Preliminaries on some basic definitions, lemmas and notations. Sect. 3 is focused upon solvability of the problem (1). Sect. 4 is devoted to approximate controllability of the system (1).

2.
Preliminaries. This section is mainly concentrated on some basic facts which are used throughout this paper.
For a given Hilbert space H, L(H) denotes the space of all bounded linear operators from H into itself with uniform operator norm, and C(I, H) is the Banach space of all continuous functions from I to H with sup-norm. For p ∈ [1, +∞), the space L p (I, H) is a set formed by all p-th H-valued Bochner integrable functions on Let Z be a Banach space with its dual Z * , and the symbol ·, · stands for the duality pairing between Z and Z * . The notation P(Z) represents a class of nonempty subsets of Z. Denote by P b (Z) = {Ω ∈ P(Z) : Ω is bounded}, and P cp,cv (Z) = {Ω ∈ P(Z) : Ω is compact and convex}.
A multi-valued map G : Also, G is called to be completely continuous if G(B) is relatively compact for every B ∈ P b (Z). If G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., u n → u * , g n → g * , g n ∈ G(u n ) imply g * ∈ G(u * ).
A multi-valued map : I → P(Z) is said to be measurable if −1 (C) = {t ∈ I : (t) ∩ C = ∅} ∈ Σ for each closed set C ⊆ Z. If : I × Z → P(Z), then measurability of means that −1 (C) ∈ Σ ⊗ B Z , where Σ ⊗ B Z is the σ-algebra of subsets in I × Z generated by the sets A × B, A ∈ Σ, B ∈ B Z , and B Z is the σ-algebra of the Borel sets in Z.
The Clarke directional derivative h 0 (x, d), of a locally Lipshitz function h : Z → R at x in the direction d, is given by The Clarke sub-differential ∂h(x), of h at x, is a subset of Z * given by We list some fundamental properties below.
Definition 2.2. [21,38] An operator valued function R(t, s) ∈ L(H) for t, s ∈ I is called to be the resolvent operator of the equation if it satisfies the following properties: (1) R(t, s) is strongly continuous in s and t, R(t, t) = I (the identity operator on H). And there exist some constants M, β such that R(t, s) ≤ M e β(t−s) , t, s ∈ I.
The following facts are also crucial in proving our main results.

Lemma 2.3. [14]
Let Z be a Banach space. If Ω ⊂ Z is nonempty, close and convex with 0 ∈ Ω, and F : Ω → P cp,cv (Ω) a u.s.c. multifunction that maps bounded sets into relatively compact sets, then one of the following statements is true:

Lemma 2.4. [14]
If D is a compact subset of a Banach space Z, then its convex closure conv(D) is compact.
3. Existence of mild solutions. In order to obtain the approximate controllability result, it is needed to investigate the solvability of the problem (1). We first impose the following conditions on the problem (1).
Lemma 3.1. If the condition (A3) holds, then for u ∈ L 2 (I, H), the set S(u) has nonempty, convex, and weakly compact values.
Lemma 3.2. Assume that the condition (A3) holds. Let the operator S satisfy: z n → z in L 2 (I, H), w n ∈ S(z n ), and w n → w weakly in L 2 (I, H), then we have w ∈ S(z).
In view of the assumption (A2), we can see that From spectrum theory of operators, we know that exists and bounded, where I is the identity operator. Moreover, the above defined operator ∆ can be expressed by Neumann expression For any g(t) ∈ C(I, H), we know the mild solution to the following problem can be expressed by (see [21,38]) By (4), we have In view of the nonlocal initial condition u(0) = m k=1 c k u(t k ) and (5), we obtain Considering (A2) and the definition of ∆, we further have Now taking (7) into (4), we obtain which gives a mild solution to the following linear problem with non-local initial conditions In convenience, we introduce the following notion known as a Green function where (8) and (10), we deduce that the mild solution to (9) can be expressed simply by

Now by
Through the previous preparation, we have the following definition.
Given v ∈ L 2 (I, V), a function u ∈ C(I, H) is called to be a mild solution to the problem (1) if there exists a function g ∈ L 2 (I, H) such that g(t) ∈ ∂G(t, u(t)) a.e. t ∈ I and the following equation holds: For each r > 0, we define B r := {u ∈ C(I, H) : u(t) ≤ r, t ∈ I}. The existence result for the problem (1) can be stated as following. where Proof.
has nonempty values, where Γ(t, s) is given by (10). From Definition 3.3, seeking a mild solution to (1) is equivalent to finding a fixed point of F defined by (13). In the following, we shall prove that the operator F admits a fixed point via Lemma 2.3. Firstly, for each u ∈ C(I, H), F (u) is convex by the convexity of S(u). For the seek of convenience, the remainder of the proof is divided into several steps.
Step 2. F is equicontinuous on B r .
For each u ∈ B r , and f ∈ F (u), there exists g ∈ S(u) such that (13) holds for t ∈ I. Take 0 ≤ t < t ≤ b. Observe that

Now taking into account that
and the compactness of R(t, s)(t − s > 0) (see (A1)) implies the continuity of R(t, s) in the uniform operator topology, we conclude by the Lebesgue dominated convergence theorem that I 3 → 0 as t → t and δ → 0.
As a result of above arguments, f (t ) − f (t ) approaches zero independently of u ∈ B r as t − t → 0. Thus, F is equicontinous on B r .
At first, for t = 0, Ω(0) := {(F u)(0) : u ∈ B r }, we see that Let δ be a real number satisfying 0 < δ < t k , we further introduce The compactness of R(t, s)(t − s > 0) implies that the set is compact for all δ > 0. Then conv(O δ ) is also a compact set according to Lemma 2.4. By mean value theorem for Bochner integrals, we have f δ (0) ∈ (t k −δ)conv(O δ ) for all t ∈ I. Thus the set Thus, there are relatively compact sets Ω δ (0) arbitrarily close to the set Ω(0), and Ω(0) is also relatively compact in H. Next, let 0 < t ≤ b and η be a real number satisfying 0 < η < t, we introduce By the compactness of R(t, s)(t − s > 0), the set is compact for all η > 0. It is again by Lemma 2.4 that the set conv(Q δ ) is also compact. From mean value theorem for Bochner integrals, we have f η 2 (t) ∈ (t − η)conv(Q δ ) for all t ∈ I. Therefore, the set Hence, there are relatively compact sets Ω η 2 (t) arbitrarily close to the set and Ω 2 (t) is also relatively compact in H. Together with the compactness of R(t, 0)(t > 0), the set Ω(t) := {(F u)(t) : u ∈ B r } is relatively compact in H for all t ∈ (0, b]. From all above arguments, we conclude that the set Ω(t) := {(F u)(t) : u ∈ B r } is relatively compact in H for every t ∈ I.
Step 4. F has a closed graph.
Let u n → u * in C(I, H) and f n → f * in C(I, H) with f n ∈ F (u n ). We need to show that f * ∈ F (u * ). The fact f n ∈ F (u n ) implies that there exists g n ∈ S(u n ) satisfying R(t k , s)g n (s)ds + t 0 R(t, s)g n (s)ds. (14) From (A3), we know that {g n } n≥1 ⊆ L 2 (I, H) is bounded. Thus, we may assume, passing to a subsequence if necessary, that g n → g * weakly in L 2 (I, H).
From (14), (15) and the compactness of R(t, s)(t − s > 0), we have By Lemma 3.2 and (15), we know g * ∈ S(u * ). Considering (16), and f n → f * in C(I, H) with f n ∈ F (u n ), we can infer that f * ∈ F (u * ). By Steps 1-4 together with Arzelà-Ascoli theorem, we conclude that the multivalued map F is completely continuous, u.s.c. with convex values. Conducted as the proof of Step 1, it is easily shown that the assertion (i) in Lemma 2.3 is not true. Thus, F admits a fixed point u in B r , which in turn is a mild solution to the problem (1). The whole proof is finished.
4. Approximate controllability. In this section, we mainly establish the approximate controllability of the system (1). Define the set is a mild solution to system (1) related to a control v ∈ L 2 (I, V)}, which is known as the reachable set of system (1) at terminal time b. Denote by R b (G) the closure of R b (G) in H. Let E be the finite dimensional subspace of H, and P E is the orthogonal projection from H to E. We introduce the following definitions (see also [23,28]).  From Definition 4.2, we can see that the control v can be selected such that the final state u (b) is approximately controllable (i.e. the condition (a) holds), and satisfies simultaneously a finite number of exact constraints (i.e. the condition (b) holds). We fist consider the following linear system related to (1).
Let us introduce the following operators defined on the Hilbert space H by where Γ * (b, s) = m k=1 χ t k (s)R * (b, 0)∆ * R * (t k , s) + χ t (s)R * (b, s), s ∈ I, and R * , ∆ * , B * denote the adjoint operators of R, ∆, B, respectively. Let u v be a mild solution to the system (17) corresponding to the control v ∈ L 2 (I, V). Then the system (17) is said to be approximately controllable on I, if for every desired final state u b ∈ H and > 0, there exists a control v ∈ L 2 (I, V) such that u v (b) − u b ≤ . Next, we give a useful characterization of the approximate controllability for (17) in Green function terms. Define the functional Lemma 4.3. For given > 0 and u b ∈ H, the above defined functional (19) admits a unique optimal control v (·) ∈ L 2 (I, V) such that Clearly, the functional L is convex and differentiable. Thus, it attains a minimum v which satisfies L (v ) = 0. Then, for all ρ ∈ L 2 (I, V), we have Owing to the arbitrariness of ρ ∈ L 2 (I, V), we obtain that for a.e.
Therefore, we further have The existence and uniqueness of an optimal control can be deduced from the general theorem on linear regulator problem (see [10]).
From above lemma, we can easily deduce the following result.
Lemma 4.4. The linear system (17) corresponding to the system (1) is approximately controllable on I if R( , Π b 0 ) → 0 as → 0 + in the strong operator topology. Therefore, we assume further that (A5) R( , Π b 0 ) → 0 as → 0 + in the strong operator topology. Now we shall prove the approximate controllability of the system (1). Proof. For all > 0 and u b ∈ H, by Theorem 3.4, the problem (1) has at least one mild solution u ∈ B r on I. Thus, in view of Lemma 4.3, there exists g ∈ S(u ) such that with and Through (10), (18) and (20)- (22), we can easily deduce that It follows from (A3) that b 0 g (s) 2 ds We see that {g ε } is bounded in L 2 (I, H), and there exists a subsequence converging weakly to g ∈ L 2 (I, H). We can know that Γ(t, s)(t − s > 0) is also compact by the compactness of R(t, s)(t − s > 0) and (10). Let From (22), (24) and the compactness of Γ(t, s)(t − s > 0), we have From (23), (25) and (A5), it follows that ) − W → 0, as → 0 + . In view of Lemma 4.4 and above arguments, the problem (1) is approximately controllable on I. This ends of proof.
In the final, we investigate the finite approximate controllability of the system (1). We define the following functional where > 0, u ∈ B r , Ψ : H → H, and with g ∈ S(u) and u b ∈ H. We now have the following fact.
Lemma 4.6. Assume that condition (A1)-(A5) are satisfied. Then, for any ball B r , there exists ε = R( , Π b 0 ) such that the functional L given by (26) satisfying Proof. The proof can be conducted similarly as [28,Lemma 3]. If the assertion of the lemma is not true, then for any ε > 0, there exist sequences {Ψ n } ⊂ H, {u n } ⊂ B r with Ψ n → ∞, such that lim n→∞ L (Ψ n ; u n ) Ψ n < ε. As Step 3 in the proof of Theorem 3.4, we can see that the set Q := {p(u(·)) : u ∈ B r } is relatively compact in H. Thus, without loss of generality, for some p(u(·)), we may assume by selection of a subsequence if necessary that p(u n (·)) → p(u(·)), strongly in H.
Considering that E is finite dimensional and P E is compact, we have P EΨn → 0 in H and ( , which contradicts (28). The proof is complete. Thus for any Φ ∈ H, we have In view of the definition of p(u(·)) and u(t), we have Therefore, The proof is complete.