Approximate controllability for some integrodifferential evolution equations with nonlocal conditions

The main objective of this work is to investigate the existence of mild solutions and approximate controllability for some integrodifferential evolution equations with nonlocal conditions. Assuming that the linear part is exactly null and approximately controllable, using resolvent operator theory, we provide our main results. An example is given to illustrate the basic results of this work.


INTRODUCTION
Controllability is one of the most important fundamental concepts of mathematical control theory, which plays an important role in deterministic and stochastic systems (see Trélat [1] and the references given there).Controllability of partial differential equations is a developing topic.Its history began with the case of the finite dimension, its extension to the infinite dimension has known a very important development since the works of Hector Fattorini in 1971, David Russell in 1978, and Jacques-Louis Lions in the late 1970s [2,3].
In an abstract way, a control system given in a space of states functions X and a space of control functions U by the following evolution law: { x ′ (t) =  (t, x(t), u(t)) for t ∈ [0, a] x(0) = x 0 where x(t, x 0 , u) ∈ X is the state of the system at time t corresponding to the initial state x 0 and the control function u(•), with a given function  , is said to be controllable if it can be brought in finite time a, from an arbitrary initial state x 0 to a prescribed final state x 1 under the action of a force u ∶ [0, a] → U.If there exists a function u such that x(a, x 0 , u) = x 1 , we say that the system is exactly controllable on [0, a].Triggiani [4] explained that the concept of exact controllability is generally stronger in spaces of infinite dimension.Then, it is more practical to look for a weaker concept of controllability known as approximate controllability.The approximate controllability ensures that it is possible to control a movement from any point to an arbitrary vicinity of any other point, but generally, the trajectory never reaches the given end point [4,5].In the past few years, some sources have reviewed the issue of approximate controllability under different conditions, that we can't confine them to a one or two.Mahmudov [6] studied the approximate controllability results of the following equation: { x ′ (t) = Ax(t) +  (x(t)) + Bu(t) for t ∈ [0, a] x(0) = x 0 + g(x), in a separable Hilbert space .The approach given by the author based on the fact that  is  1 with Fréchet derivative, and there exists L > 0 such that || ′ (z)|| () ≤ L for each z ∈ , where g is a Lipschitz continuous function, which satisfying the following condition: ∃ L g > 0 such that ||g(z)|| ≤ L g (||z|| ∞ + 1) for all z ∈ ([0, a]; ).Partial ( or ordinary) integrodifferential equations are very often used as mathematical models to describe several natural phenomena such as physical, biological and engineering sciences, and so on.Volterra [7] suggested that the dynamics of certain types of elastic materials can be take the following form of partial integrodifferential equations: where  and  are appropriates functions.Davis [8] and Bloom [9] (pp 101-104), mentioned that the electric displacement field in Maxwell Hopkinson dielectric must satisfy the following system of partial integrodifferential equations: for T > 0 and Ω ⊂ R 3 , where ,  ∈ R and  is a vector of scalar functions.An other example of reaction diffusion equation is given in Section 5 as an illustration of how this work could be put to use.We refer the readers to previous studies [7][8][9][10][11] and the references given there for more examples of applications.The importance of studying the controllability of this type of equations is not less important than that of classical differential equations.The literature is rich and it is not the place here to recall the large number of issues devoted to this purpose.On the other hand, the notion of nonlocal condition has been introduced by Byszewski [12] when he studied the existence and the uniqueness of solutions of nonlocal Cauchy problems.As mentioned by Byszewski [13,14] and Deng [15], the motivation come from physics, since the nonlocal conditions can be more practical and with a better effect than the classical initial conditions to describe some physical phenomena.Day [16,17] proposed a linear parabolic equation with nonlocal boundary condition arising from static thermoelasticity.Kerefov [18] and Vabishchevich [19] considered a one-dimensional parabolic equation with nonlocal initial condition to study the dynamic of gas in transparent tube.As an example, the nonlocal function can be take the following form: In this work, we study the approximate controllability of the following integrodifferential evolution equation: ( which is a generalization of the approach given by Mahmudov [6] since G ≠ 0 and  is a nonautonomous function. More precisely, X equipped with the norm || • || is an infinite dimension reflexive Banach space, and X * its dual equipped with the norm || • || * .Here,  ), and (Y , X) denotes the space of all bounded linear operators defined from Y to X.We denote by (Z) the space of bounded linear operators defined on a Banach space Z.Throughout this paper,  (•, x) is strongly measurable for each x ∈ X.
Grimmer [20] considered the following integrodifferential equation: where h ∈ L 1 loc (R + ; X).Using resolvent operator theory, the author obtained a variation of constant formula which enable to obtain some results concerning existence, regularity and asymptotic behavior of solutions for Equation (2).Recently, Ezzinbi and Ghnimi [21] studied the nonlocal problem (1).They supposed that the linear part has a resolvent operator in the sens given by Grimmer and the nonlinear part is assumed to be continuous and Lipschitzian with respect to the second argument.In this note, we study the approximate controllability for Equation (1) in a reflexive Banach space, with G ≠ 0, and  satisfied the condition of uniform boundedness of the Fréchet derivative with respect to the second argument, that is, there exists The following is how this paper is organized.In Section 2, we start by introducing some important results concerning theory of resolvent operators in the sense given by Grimmer.Some definitions, some results concerning fixed point theory, and some used results are given.In Section 3, using Sadovskii fixed point theorem, we present a result concerning the existence of mild solution for Equation (1).In Section 4, assuming that the linear part corresponding to Equation ( 1) is approximately and exactly null controllable on [0, a], we present our main controllability results for Equation (1).In the end, an example is given to illustrate our theoretical approaches.

PRELIMINARIES
Consider the following integrodifferential equation: We assume the following assumptions: (H 1 ) A generates a strongly continuous semigroup (T(t)) t≥0 on X. (H 2 ) For all t ≥ 0, G(t) ∈ (Y , X), and for each x ∈ Y , the function G(•)x is bounded, differentiable and the derivative G ′ (•)x is bounded and uniformly continuous on R + .

Definition 1.
A family (R(t)) t≥0 ⊂ (X) is called to be a resolvent operator family of Equation (3) if the following conditions are verified: 1. R(0) = I and ||R(t)|| (X) ≤ Me t for some constants M ≥ 1, and The next theorem states a result concerning existence of the resolvent operator for Equation (3).
We denote by l() the Laplace transforms of a given function l(t).The following theorem will be needed to give an explicit form to (R(t)) t≥0 .
Theorem 2 (theorem 3.1 in Grimmer and Kappel [23]).Assume that the following conditions are satisfied: a) A generates an analytic semigroup (T(t)) t≥0 and satisfies the following estimate: x is strongly measurable for each x ∈ Y. c) For  ∈ C with Re() > 0, Ĝ() exists as an element of (Y , X) and
Theorem 3 (Liang et al. [24]).Assume that (H 1 ) and (H 2 ) are satisfied.Let (R(t)) t≥0 be the resolvent operator of Equation (3).Then, 1.For any  > 0, there exists a positive constant C ≡ C() such that Let consider the following linear system: The controllability results of Equation ( 4) has been studied in many works (see, e.g., Tucsnak and Weiss [25] and the references given there).Let La 0 ∶ L 2 ([0, a]; U) → X be the controllability map given by where (T(t)) t≥0 is the C 0 -semigroup generated by A. Under some conditions on the map La 0 , the authors established the exact, the exact null, and the approximate controllability for Equation (4).
In this paper, Im(P) stands as the range of a given map P. We recall the following theorem.
(i) Equation ( 4) is exactly controllable on [0, a] if and only if Im( La 0 ) = X.(ii) Equation ( 4) is approximately controllable on [0, a] if and only if Im( La 0 ) = X.(iii) Equation ( 4) is exactly null controllable on [0, a] if and only if Im(T(a)) ⊆ Im( La 0 )).By this motivation, we recall some controllability concepts of the following equation: (5) Since, t → Bu(t) ∈ L loc (R + , X), it follows that Equation (5) has a unique mild solution given by Let denote by L a 0 the map defined from L 2 ([0, a]; U) to X by Definition 5. We say that the control system ( 5) is exactly controllable on [0, a] if Im(L a 0 ) = X.
Proposition 1. Assume that R(t)B is compact for every t > 0.Then, system ( 5) is not exactly controllable on [0, a].
Proof.Using the fact that R(t)B is compact, we show that L a 0 is compact [26].Let define By contradiction, we assume that the control system ( 5) is exactly controllable on [0, a].Then, La 0 is bijective and compact, which implies that X is a finite dimension space.That is a contradiction.□ Corollary 1. Assume that T(t) is compact for every t > 0.Then, system ( 5) is not exactly controllable on [0, a].Definition 6.The system ( 5) is said to be exactly null controllable on [0, a] if for all x 0 ∈ X, there exists where x(a, x 0 , u) is the state value of Equation ( 5) at terminal time a corresponding to the control function u(•), and the initial value x 0 .
Proposition 2. The system ( 5) is exactly null controllable on [0, a] if and only if and the control function u Proof.Assume that system ( 5) is exactly null controllable on [0, a], then there is which implies that system ( 5) is exactly null controllable on [0, a].□ Definition 7. We say that the control system ( 5) is approximately controllable on ) is dense in X and Equation ( 5) is exactly null controllable on [0, a], then Equation ( 5) is approximately controllable on [0, a].
Let J ∶ X → X * be the duality mapping given by Lemma 2 (Barbu and Precupanu [30]).The mapping J is bijective, demicontinuous (i.e., continuous from X with a strong topology into X * with weak topology ), and strictly monotonic.Moreover, J −1 ∶ X * → X is also a duality mapping.
We denote by Δ ∶= {(t, s); Proposition 3. Let (G t ) t≥0 be a family in ([0, a], (X)).Assume that there exists a bounded function Then, there exists a unique family {P(t, s, G t ), (t, s) ∈ Δ} ⊂ (X) such that Proof.Let x ∈ X be fixed.Let denote by (Δ; X) the space of continuous functions defined from Δ to X equipped with the norm We define the map  x on (Δ; X) by We show that  x has a unique fixed point in (Δ; X).In fact, let z 1 , z 2 ∈ (Δ; X), then Hence, Thus, Let n ≥ 1 be sufficiently large such that This finish the proof.
where P(a, s, G) * is the adjoint operator of P(a, s, G) defined on X * by The following lemma is needed in this paper.
Lemma 3. Assume that system ( 5) is approximately controllable on [0, a].Then, for every  > 0, and for any G ∈ ([0, a], (X)), the map I +  a 0 (G)J is a bijection, and Proof.Let G ∈ ([0, a], (X)), and x * ∈ X * .Then, Since J is strictly monotonic, Let  ∈ X.We show that there exists a unique z ∈ X such that Let X 0 be the subspace of X introduced by The spaces X 0 and X * 0 are strictly convex, separable, and reflexive [29].Let J 0 = J | X 0 .We define the map  ∶ X * 0 → X 0 by Then, The Minty-Browder theorem (see theorem 2.2 in Showalter [31]) shows that 0 ∈ Im(), that is, there exists which implies that ||z|| ≤ ||||.Now, we prove that then we can extract a subsequence of ( J(z  )) >0 that we will continue to denote by the same index  > 0, such that ⟨J(z  ), z⟩ → ⟨z * , z⟩ as  → 0 + for every z ∈ X and for some z * ∈ X * .Using the fact that J is bijective, we can find a z ∈ X such that J(z) = z * , and ⟨J(z  ), z⟩ → ⟨J(z), z⟩as → 0 + for every z ∈ X.
Let E be a Banach space.For an arbitrary bounded set Ω ⊂ E, we introduce the notion of the Hausdorff measure of noncompactness.See Kamenskii et al. [33] for more details.
Definition 10.The Hausdorff measure of noncompactness of the set Ω is defined by Lemma 6 (Kamenskii et al. [33]).Ω is relatively compact if and only if (Ω) = 0.
Definition 11.An operator Φ defined from a Banach space E to a Banach space F is called condensing (−condensing) if it is continuous and for every bounded noncompact set Ω ⊂ E the following inequality is satisfied:

𝛼(Φ(Ω)) < 𝛼(Ω).
For example, contractions operators and completely continuous operators are condensing operators.Now, we are in a position to state the Sadovskii fixed point theorem.
Theorem 5 (Sadovskii fixed point theorem [32]).Let Φ ∶ C ⊂ E → E be a condensing map.Assume that C is convex, closed and bounded.If Φ(C) ⊆ C, then Φ has at least one fixed point in C.
Our basic assumptions in this work are the following: for allx ∈ X, and t ∈ [0, a].

EXISTENCE OF THE MILD SOLUTION FOR EQUATION (1)
Let u(•) ∈ L 2 ([0, a]; X) and x 0 ∈ X be fixed.We define the operator Remark 2. The operator Q(•, u, x 0 ) satisfies the following equation: By Fubini's theorem, we can affirm that In the next, we use the shorthand notation Q(•) instead of Q(•, u, x 0 ).Let Φ be the map defined on ([0, a]; X) by for each x ∈ ([0, a]; X), and t ∈ [0, a].Remark 3. A fixed point of operator Φ is a mild solution of Equation (1).
Indeed, let x ∈ ([0, a]; X) be a fixed point of Φ.Then, We recall that The following theorem is the main result in this section.The theorem ensures the existence of such a mild solution of Equation ( 1).
Theorem 6. Assume that (H 3 ) -(H 5 ) hold.If M a M g < 1, then the map Φ has a fixed point on ([0, a]; X), which is a mild solution of Equation (1).
Proof.Firstly, we show that there is r > 0 such that Φ( B(0, r)) ⊆ B(0, r), where B(0, r) is the closed ball in ([0, a]; X) with center at 0 and radius r >.In fact, if we assume that this assertion is false, that is, for every r > 0 there exists We divide by r, we obtain that Letting r → +∞, we get that 1 ≤ M a M g , which is a contradiction to our assumption.Thus, there is r > 0 such that Since the assumption (H 5 ), we can affirm that As (F t (•)) t∈[0,a] are continuous, using Gronwall's lemma, and the dominated convergence theorem, we get the desired continuity of Φ.Moreover, we have (Φ(Ω)) ≤ (Ω) for any bounded set Ω of ([0, a]; X).Indeed, let Ω be a bounded set of ([0, a]; X).Let x ∈ Ω and 0 ≤ t 1 < t 2 ≤ a.Then, Since R(t) is norm-continuous, using dominated convergence theorem, we can see that It remains to prove that V(t) is relatively compact in X.Let 0 < t ≤ a, and 0 <  < t.For x ∈ Ω, we define where B denotes the open ball in X with center at x i and radius  2 .On the other hand, By dominated convergence theorem and Theorem 3, we obtain that ||Q(x)(t) − Q  (x)(t)|| → 0 as  → 0.Then, Hence, V(t) is also relatively compact in X.Thus, Q is relatively compact which implies that (Q(Ω)) = 0. On another side and for x,  ∈ Ω, we have Using Theorem 5, we deduce that Φ has a fixed point in B(0, r).□

APPROXIMATE CONTROLABILITY OF EQUATION (1)
Let d ∈ X, x 0 ∈ X and  > 0. We define the map Q  (•, u, x 0 ) on ([0, a]; X) by where It is immediate to see that Q  (•, u, x 0 ) has the same property in Remark 2.
Proof.Let  > 0, t ∈ [0, a], and x ∈ ([0, a]; X).Then, Then, Since w(x n , F a ) → w(x, F a ) strongly in X, using (2), and (4) from Lemma 4, we can affirm that lim Now, H be the map defined on ([0, a]; X) by x is a mild solution of Equation ( 1) with the control function Indeed, let  > 0 and x is be a fixed point of Φ  .Then, The following theorem is the main result in this section.The result is a generalization of the main result given by Mahmudov [6] in the nonautonomous case, when G ≠ 0 and then (R(t) t≥0 ) is not a C 0 -semigroup on X.We recall that M g comes from assumption (H 5 ),  comes from assumption (H 7 ) and proposition 2. M a and M B are given by Proof.Let  > 0. Firstly, we show that there is r > 0 such that Φ  ( B(0, r)) ⊆ B(0, r), where B(0, r) is defined as in the proof of Theorem 6.In fact, if we assume that this assertion is false, that is, for every r > 0 there exists x r ∈ B(0, r), Using assumption (H 4 ), we can affirm that we divide by r, we obtain that , which is a contradiction to our assumption.Thus, there exists On the other hand, we have As u  (s, .), and F t (•) are continuous, using Gronwall's lemma, we get that Q  is continuous.Thus, Φ  itself is continuous.Moreover, we have (Φ  (Ω)) ≤ (Ω) for any bounded set Ω of ([0, a]; X).Indeed, let Ω be a bounded set of ([0, a]; X).Let x ∈ Ω and 0 ≤ t 1 < t 2 ≤ a.Then, Since R(t) is norm-continuous, using dominated convergence theorem, we can see that According to the previous proof of Theorem 6, we can affirm that Q  (Ω)(t) is relatively compact.By Arzela-Ascoli theorem, we infer that Q  (Ω) is relatively compact, which implies that (Q  (Ω)) = 0.

APPLICATION
To illustrate our theory, we consider the following control system of reaction diffusion equation: where  = [0, a] × (0, ), x 0 ∈ L 2 (0, ),  > 0, (c i ) 1≤i≤p is a finite sequence of a given real numbers.The function q(•, ) is strongly measurable for each  ∈ R, and q is  1 with respect to the second argument.Moreover, there exists Assume that Λ(•) is a bounded differentiable function on R + with a bounded uniformly continuous derivative Λ ′ (•).
Assume that there exists N ≥ 1 and  > 0 such that the Laplace transform Λ(•) of Λ(•) satisfy the following condition: Let X = L 2 (0, ) be the space of square integral function from (0, ) to R endowed with the metric ||.||L 2 = ||.||L 2 (0,) induced by the following inner product: where  and g are square integral functions.L 2 (0, ) is a separable Hilbert space.
Define the operator It is well known that A generates a strongly continuous semigroup (T(t)) t≥0 , then (H 1 ) is satisfied.Moreover, T(t) is compact for each t > 0, then (H_3) is satisfied, and system (6) is not exactly controllable on [0, a].
Let G(t) ∶ D(A) ⊂ X → X be the operator defined by G(t)z = Λ(t)Az for t ≥ 0 and z ∈ D(A).
In addition, Using the fact that q is  1 with respect to the second argument, we show that Then, each subsequence of (Θ(v n )) n≥0 has a sub-subsequence which converges to Θ(0).Hence Θ(v n ) → Θ(0), and Θ is continuous at 0. Using theorem 3.12 from Appell and Zabreiko [35], we can show that  (t, .) is differentiable at w ∈ X, and  is  1 with respect to the second argument.Moreover,    (t, z)() =   q(t, z()) for a.e s ∈ (0, ), and z ∈ X, which implies that ||   (t, z)|| (X) ≤ l q (t), then (H_4) holds.We can also show that (H_5) is satisfied with To give an explicit form to R(t), we use the result given by Theorem 2. Firstly, we show that conditions for some constant  ∈ R + such that   > 6N.Hence, .
By a direct calculation, we get that .□ Proposition 5.The linear system corresponding to Equation ( 6) is approximately controllable on [0, a] for every a > 0.
Proof.Proof.The proof is a direct application of Theorem 7. □