Controllability in Dynamics of Diffusion Processes with Nonlocal Conditions

The paper deals with semilinear evolution equations in Banach spaces. By means of linear control terms, the controllability problem is investigated and the solutions satisfy suitable nonlocal conditions. The Cauchy multi-point condition and the mean value condition are included in the present discussion. The final configuration is always achieved with a control with minimum norm. The results make use of fixed point techniques; two different approaches are proposed, depending on the use of norm or weak topology in the state space. The discussion is completed with some applications to dynamics of diffusion processes.


Introduction
This paper concerns the second order integro-differential equation for x ∈ D ⊂ R n bounded and with a sufficiently regular boundary and t ∈ [0, T ] := J which is a model for the description of diffusive behaviours such as the propagation of electro-magnetic waves, the motion of a string or a membrane with external damping, the evolution of visco-elastic fluids and the heat propagation (see e.g., [14,18,19] and the references therein). The space domain D is a bounded subset of the Euclidean space R n , n ≥ 1. The nonlocal term in integral form appearing in (1) accounts of longdistance interactions into the process (see e.g., [13,15]). As usual, the Laplace operator stands for a diffusion behaviour of punctual type. The nonlinear part f (t, x, z, Ω h(x, ξ)z(t, ξ) dξ) can also be replaced by an interval such as, for instance, D h(x, ξ)z(t, ξ) dξ . (2) This case occurs, for instance, when the function f in (1) displays some jump discontinuity or it is known up to some degree of uncertainty; in this case Eq. (1) turns to a multivalued dynamic.
With v(t, x) given, by a solution of (1) we mean a function z : J × D → R such that z t (t, x) exists, for t ∈ J and a.a. x ∈ D, z(t, ·), z t (t, ·) ∈ L 2 (D, R), t ∈ J and the map y : J → L 2 (D, R) defined by y(t) = z(t, ·) belongs to C 1 (J; L 2 (D, R)). The symbol A stands for the set of functions z(t, x) with all previous properties. Equation (1) is then satisfied in integral form (see Sect. 6 for details).
We are interested in solutions of (1) which satisfy some nonlocal conditions. Consider, for instance, the Cauchy multi-point condition where and z 0 (·), z 0 (·) are suitable real valued functions. A second important example of nonlocal condition is the weighted mean value condition z(0, x) = z 0 (x) + T 0 k 1 (t)z(t, x) dt, z t (0, x) = z 0 (x) with z 0 (·), z 0 (·) as in (3) and k i (·), i = 1, 2 suitable real valued functions. The initial value problem associated to (1), i.e., z(0, ·) = z 0 (·) and z t (0, ·) = z 0 (·) is clearly included, in both case. The class of nonlocal conditions that we are able to manage is, indeed, quite wide (see Definition 1.1). The additional term v(t, x) appearing in (1) accounts of external forces acting into the model, i.e., it is a control of linear type. The main aim of this paper is to investigate the possibility to act by v(t, x) for obtaining a solution of (1) which satisfies some given nonlocal condition (for instance, (3) or (5)) and reaches a prescribed configuration at time t = T , i.e., z(T, ·) = z 1 (·), where z 1 (·) stands for a suitable real valued function. This is known as the exact controllability problem associated to (1) (for short controllability problem in the sequel). The possibility to lead a system in finite time into a desired configuration is of great interest in several physical applications.

Controllability in Dynamics of Diffusion Processes
Page 3 of 28 78 Definition 1.1. We say that problem (1)- (6) is controllable if, for every z 0 (·), z 0 (·), z 1 (·) ∈ L 2 (D, R), there is a control function v(t, x) and a corresponding solution z(t, x) of (1) satisfying both (6) and As usual in this framework we transform problem (1)-(6) into its abstract setting. When assuming that the nonlinearity in (1) takes the form as in (2) and the linear term also depends on t, we arrive to the multivalued nonlocal probleṁ The discussion about Problem (8) and (9) is presented in Sect. 3. We give there the notion of solution for (8) and (9) (Definition 3.1) and define its controllability (Definition 3.2). The main results of the paper are Theorems 3.1 and 3.2; they provide sufficient conditions for the controllability of (8) and (9). It was pointed out by Triggiani [24] that, in infinite dimensional Banach spaces, the compactness of the associated evolution operator is in contradiction with the controllability of a linear system while using locally L p -controls, for p > 1. We overcome this lack of compactness by means of two different strategies; precisely by introducing suitable measures of noncompactness in Theorems 3.1 and by making use of the weak topology of E in Theorem 3.2. Fixed point techniques are used in both cases in the proofs. The final configuration is always achieved with a control with minimum norm in L 2 (J, U). The proofs of Theorems 3.1 and 3.2 are, respectively, contained in Sects. 4 and 5.
The controllability in infinite dimensional setting with M ≡ 0 was recently treated in [5,20]; previous contributions are reported in [9] and in the survey [2]. A Cauchy multi-point condition with t k fixed in J and given real values c k , k = 1, . . . , p was introduced in [4] (see also [3]); the linear part there takes the form A = A(t, y), but A(t, y) is a bounded operator.
The notion of approximate controllability (see e.g., [23]) seems the most appropriate in the case of a compact evolution operator. Section 6 deals with the controllability problem associated to Eq. (1); for the sake of simplicity we restrict there to the case when x is a real variable and D an interval and we consider the multivalued Cauchy condition (3). This section also contains a discussion concerning the cases when the use of Theorem 3.1 is preferable than Theorem 3.2 and vice versa. The methods used in Sect. 3 in abstract setting is quite general and hence it can be used also for the study of some first order integro-differential models such as In Sect. 7 we discuss the controllability of (10). At last, Sect. 2 contains some basic notation and results concerning multivalued analysis and fixed point theory and some relevant example of measures of noncompactness.

Preliminaries
In this section we briefly introduce the theory of multivalued analysis, show some relevant examples of measures of noncompactness (m.n.c. for short) and discuss their main properties, recall some useful function spaces and the fixed point results used in the following. We denote by X or Y a topological space and by E or F an arbitrary Banach space with norm · . We start with the introduction of some definitions, notation and preliminary items from multivalued analysis and linear operators.
A multivalued mapping (multimap for short) φ : X Y is a relation that assigns to any point x ∈ X a nonempty closed set φ is sequentially closed when the conditions lim n→+∞ x n = x 0 , lim n→+∞ y n = y 0 , and y n ∈ φ(x n ), n ∈ N, imply that The multimap φ is upper semicontinuous (u.s.c. for short) if the set We say that φ has a fixed point if X ⊆ Y and there exists x ∈ X such that x ∈ φ(x).
Given a sequence {x n } ⊂ E we write x n → x 0 and x n x 0 , with x 0 ∈ E, for denoting, respectively, the strong and weak convergence in E.
As usual C([a, b], E) is the Banach space of continuous functions f : [a, b] → E with norm · C . The weak convergence in this space is discussed in the following lemma   By L(E, F) we denote the Banach space of automorphism (i.e., linear and continuous operators) T : E → F with norm T L := sup x ≤1 T (x) . We say that the automorphism T ∈ L(E, F) has a right inverse, if there exists a linear continuous operatorT : F → E such that T •T is the identity on F. The right inverseT is called a pseudoinverse ifT (u) = x implies that x = min{ y : T (y) = u}.
We recall now the definition of m.n.c., report its main properties and propose some relevant examples. Further information and all the proofs can be found, for instance, in [1,16]. A m.n.c. β is said to be 78 Page 6 of 28 for every a ∈ E and Ω ∈ P.
In addition, if A is a cone, a m.n.c. is: A relevant example of m.n.c. is the Hausdorff m.n.c. χ E , given by χ E enjoys all the above properties. The following relation between the Hausdorff m.n.c. and a bounded linear operator can be established.
We introduce now some important m.n.c. in the space of continuous functions. Let is such that mod C (Ω) = 0, then it is an equicontinuous family of functions. We remark that neither γ nor mod C (Ω) are regular m.n.c. The further m.n.c. in C([a, b], E) with values in (R + ) 2 : is then frequently used, where the ordering is the natural one introduced by the positive cone of R 2 and Δ(Ω) stands for the collection of all countable subsets of Ω. As a consequence of the Arzelà-Ascoli Theorem, the m.n.c. ν turns out to be regular; it is also monotone and nonsingular. The following result is useful for the computation of ν.
Then, for any in the general case and in the case of a separable Banach space E.
We briefly introduce the notion of evolution system and evolution operator and refer to [21] for further details.
operator and E a Banach space is called an evolution system if the following conditions are satisfied: s)x is continuous on Δ for every x ∈ E. Given an evolution system, we can consider the respective evolution operator U : Δ → L(E).
Since the evolution operator U is strongly continuous on the compact set Δ, by the uniform boundedness theorem there exists a constant D U such that Now we recall the Pettis measurability criterion.

Statement of the Problem
In this section we introduce the notion of controllability for system (8) and (9) and state its validity in two different cases (see Theorems 3.1 and 3.2).
We start with the discussion about A(t) and B and introduce the operator G.
We assume that Let G : L 2 (J, U) → E be the linear bounded operator defined by When L 2 (J, U) is reflexive, which is the case here because U is reflexive, the operator G has a right inverse if and only if it has a pseudoinverse (see Sect. 2 and [2, page 9]); we denote it G −1 . We assume that Conditions (A), (B), (G1), (F1) and (F2) will always be assumed. For the sake of simplicity, we will no more mention them. The solution of problem (8) and (9) with u ∈ L 2 (J, U) given, is intended in integral form, precisely Definition 3.1. Let u ∈ L 2 (J, U) be given. A function y(·) ∈ C(J, E) is called a mild solution of problem (8) and (9) if the multimap t F t, y(t) has a selection f ∈ L 1 (J, E) satisfying (8) and (9) is called controllable on J if, for any y 0 , y 1 ∈ E, there is a control u ∈ L 2 (J, U) such that the corresponding mild solution y(·) satisfies y(T ) = y 1 .

Definition 3.2. Problem
As pointed out in Introduction, the notion of controllability is classical when M ≡ 0.
i.e., when F ≡ 0 and M ≡ 0, the controllability is equivalent to the existence of a pseudoinverse of G (see e.g. [9]); this is the motivation for assumption (G1).
(ii) If y(·) is a controllable solution of (8) and (9) with control u ∈ L 2 (J, U) (see Definition 3.2), then In several preliminary results (see Sect. 4) we also need the following, quite general, growth condition on F .
We establish the controllability of problem (8) and (9) in two different sets of regularity and growth conditions on F and M , which cause the use of different techniques. In particular, when the regularities involve the norm bounded. (G2s) there exists a function g ∈ L 1 (J, R + ) such that, for every bounded set Ω ⊂ E, In order to simplify notation, we denote in the following by (Hs) this group of assumptions, i.e., When the regularities are given by means of weak topology in E we assume (Ew) E is a reflexive Banach space; (F4w) F (t, ·): E E is weakly sequentially closed for a.e. t ∈ J, i.e., if x n x 0 , y n y 0 and y n ∈ F (t, x n ) for all n ∈ N, then y 0 ∈ F (t, x 0 ); (M1w) M : C(J, E) → E is weakly sequentially continuous; (M2w) M : C(J, E) → E is bounded on bounded sets.
We need in the following to consider the Nemytskiȋ operator P F : C(J, E) L 1 (J, E) associated to F which is defined by y(t)) a.e. on J}, for all y ∈ C(J, E).
P F is well-defined, in our settings, as showed by the following lemmas.   (8) and (9). They treat both the case when F (t, ·) is strictly sublinear in its variable x (see condition (s1)(i)) and when F (t, ·) is possibly linear (see condition (s2)(i)). Their proofs appear, respectively, in Sects. 4 and 5. They exploit fixed point arguments and, hence, require the introduction of a solution operator H, which will be defined in (24). The estimates on H involve, in particular, the value D B D G D U √ T ; for simplifying notation, we put in the following Additional restrictions on M and C are required, in the case when F (t, ·) is possibly linear.

Theorem 3.2. Assume (Hw). Then Problem
Given q ∈ C(J, E), f ∈ P F (q) and y 0 , y 1 ∈ E, we define and In addition to the generalized Cauchy operator Sf (see Remark 3.1(ii)), we need a further operator between the same spaces because of the presence of a control into the model. Precisely, let S 1 : L 1 (J, E) → C(J, E) be defined by Proof. The case when A(t) = A for all t ∈ J and A generates a C 0 -semigroup is discussed in [20, Lemma 1]. We use a similar reasoning in this general framework. Let f, g ∈ L 1 (J, E); for t ∈ J we have We proved that S 1 satisfies condition (i) in Theorem 2.2. Let {f n } ⊂ L 1 (J, E) be such that {f n (t)} ⊂ K for a.a. t ∈ J and f n g in L 1 (J, E) with K compact in E. Since {f n } is semicompact, by applying Theorem 2.2 to the generalized Cauchy operator S, we have that Sf n (T ) → Sg(T ) in E. Hence, for every t ∈ J, by (23) we have i.e., S 1 f n → S 1 g in C(J, E). The proof is complete.
Given y 0 , y 1 ∈ E, we introduce the solution operator H : C(J, E) C(J, E) defined by where with u f,q defined in (21). It is then straightforward to show the following estimate for the solution operator H with x f,q ∈ H(q), q ∈ C(J, E) and C defined in (19).
Remark 3.4. Let y ∈ C(J, E) be a fixed point of H, i.e., y = x f,y for some f ∈ P F (y). It is easy to see that y is a solution to problem (8) and (9) with u f,y as in (21). Furthermore, since Hence, when H has a fixed point for every y 0 , y 1 ∈ E, then problem (8) and (9) is controllable (see Definition 3.2) and the control u f,y associated to the fixed point has minimal norm.

Proof of Theorem 3.1
In this part we investigate the controllability of problem (8) and (9) Then Since F is convex-valued, we have that Proof. We prove in the following that H is sequentially closed, i.e. that given q j , q ∈ C(J, E), y j ∈ H(q j ), y ∈ C(J, E), with j ∈ N, if q j → q, y j → y in C(J, E), while j → ∞, then y ∈ H(q). Since C(J, E) is a Banach space, it implies the closure of H. Let f j ∈ P F (q j ) be such that where u j := u fj ,qj with u fj ,qj defined in (21) and p qj (f j ) in (20), for j ∈ N.
According to the convergence of {q j } we can also find a bounded set Ω ⊂ E satisfying q j (t), q(t) ∈ Ω, for all t ∈ J, j ∈ N.
(27) Cauchy operator (Remark 3.1(ii)) we obtain that Sf j → Sf in C(J, E). In particular, By the uniqueness of the limit we obtain that z = y. Now we prove that f ∈ P F (q). In fact, by Mazur's convexity Theorem we obtain a sequencẽ such thatf j → f in L 1 (J, E) and then, up to a subsequence,f j (t) → f (t) for a.a. t ∈ J. By Remark 3.2, F (t, ·) is u.s.c. for a.a. t ∈ J. Let t ∈ J be such that where B 0 is the ball of radius 1 centered at 0 in E. Since F is compact and convex valued, this implies that f (t) ∈ F (t, q(t)). Hence f ∈ P F (q) so that the proof is complete.
The compactness of Q implies the existence of a subsequence, still denoted as the sequence, such that q j → q ∈ C(J, E). By the same reasoning as in Lemma 4.2 we can prove that {y j }, and hence H(Q), is relatively compact in C(J, E). It implies that H is quasi-compact. Since H is closed (see Then, we can find sequences {q j } ⊂ Θ, {f j } ⊂ L 1 (J, E) such that f j ∈ P F (q j ) and y j satisfies (26) for j ∈ N. By means of conditions (M2s) and (F5s) and applying Lemma 2.2, we have and Since {q j } is bounded, we can find a bounded set Ω ⊂ E such that q j (t) ∈ Ω for t ∈ J and j ∈ N. So, by (F3) and with a similar reasoning as in the proof of Lemma 4.2, we obtain that {U(t, ·)f j (·)} is integrably bounded in By Lemma 2.3, the semiadditivity of the Hausdorff m.n.c. and assumption (G2s), we can estimate Therefore, by conditions (30)-(32) and according to (26), we have Owing to (s1)(iii) and (28) we can conclude that γ({q j }) = γ({y j }) = 0. We claim that the set {y j } is relatively compact in C(J, E). In this case {y j } is equicontinuous and then mod C ({y j }) = 0; hence ν (H(Θ)) = (0, 0); by (28) also ν (Θ) = (0, 0). Θ is then relatively compact by the regularity of the m.n.c. ν and hence H is ν-condensing. To complete the proof, it remains to show the relative compactness of {y j }. Notice that y j , j ∈ N satisfies the following estimate The sequence {ω j } ⊂ L 2 (J, E) and then {ω j } ⊂ L 1 (J, E). Again by the relative compactness of {M (q j )} we get that also the set {z j } is relatively compact in E. By the continuity of G −1 the set {G −1 z j } is relatively compact in L 2 (J, U) and then also in L 1 (J, U). This implies that {ω j } is relatively compact in L 1 (J, E). At last it is easy to show that {Sω j } is relatively compact in C(J, E). The claim is proved and the proof is complete.
Proof of Theorem 3.1. Fix y 0 , y 1 ∈ E and consider the solution operator H defined in (24); let Q r ⊂ C(J, E), r ∈ N be the closed ball with radius r > 0 and centre in 0. Assume that for some r 0 > 0. By means of Lemmas 4.1-4.4 we obtain that the multimap H : Q r0 Q r0 satisfies all the assumptions of Theorem 2.4 and then H has a fixed point y ∈ Q r0 , i.e., y ∈ H(y). Since the conclusion is valid for all y 0 , y 1 ∈ E, then problem (8) and (9) is controllable (see Remark 3.4).
(p1) Assume conditions (s1) We claim that H(Q n0 ) ⊆ Q n0 for some n 0 ∈ N. We reason by contradiction and hence assume the existence of two sequences {q n }, {x n } ⊂ C(J, E) such that q n ∈ Q n , x n = x fn,qn , for some f n ∈ P F (q n ) and x n ∈ Q n for all n ∈ N. Hence there exists {t n } ⊂ J satisfying x n (t n ) > n and then, by (25), By condition (s1)(i), we have n < x n (t n ) ≤ C y 1 When dividing by n and computing lim inf, by (s1)(i)-(ii) we arrive to the following contradictory conclusion Assumption (34) is then satisfied, in this case, for every y 0 , y 1 ∈ E. (p2) Assume conditions (s2) Again we claim that H(Q n0 ) ⊆ Q n0 for some n 0 ∈ N and we reason by contradiction. We consider, in particular, the sequences {q n }, {x n }, {f n } and {t n } introduced in (p1). By (35) and (s2)(i) we obtain Dividing previous inequality by n and passing to the limit, by (s2)(ii) and (s2)(iii) we arrive to the contradictory conclusion Assumption (34) is then satisfied, in this case, for every y 0 , y 1 ∈ E. (p3) Assume conditions (s3) Given y 0 , y 1 ∈ E, consider Q L with L = L(y 0 , y 1 ) as in (s3)(ii). Take q ∈ Q L and y ∈ H(q); then y = x f,q for some f ∈ P F (q). Notice that, by (s3)(i), Therefore, by (25), (s3)(i)-(ii), we obtain implying that H(Q L ) ⊆ Q L and (34) is true also in this case for any choice of y 0 , y 1 ∈ E. The proof is complete.

Proof of Theorem 3.2
In this part we investigate the controllability of problem (8) and (9)  C(J, E) × C(J, E) (Lemma 5.1) and that H has closed values and it is weakly compact when restricted to bounded sets (Lemma 5.2). We need, in the following, the Eberlein-Šmulian theory (see e.g., [17]); it states that, in E, the relative sequential weak compactness and the sequential weak compactness are, respectively, equivalent to the relative weak compactness and the weak compactness. The proof of Theorem 3.2 completes this part. Proof. Let q j , q ∈ C(J, E), y j ∈ H(q j ), y ∈ C(J, E), with j ∈ N, be such that The result is proved if y ∈ H(q). Notice that y j satisfies (26) for some f j ∈ P F (q j ), j ∈ N. By the characterization of the weak convergence in C(J, E) (see Lemma 2.1), condition (27) is satisfied, for some bounded Ω ⊂ E. According to (F3) there exists μ Ω ∈ L 1 (J, R) such that f j (t) ≤ μ Ω (t) for a.a. t ∈ J and j ∈ N; hence, by the reflexivity of E and Theorem 2.1, there is a subsequence, still denoted as the sequence, satisfying f j f ∈ L 1 (J, E). Given φ : E → R, linear and bounded and t ∈ J, consider the operator Φ : Since Φ is clearly linear and bounded and the weak convergence: f j f is true also in L 1 ([0, t], E), we have that By the arbitrariness of φ we conclude that Consequently, since M (q j ) M(q) by (M1w), we obtain that p qj (f j ) p q (f ) as j → ∞; hence, the linearity and boundedness of G −1 imply that u j := u fj ,qj u f,q in L 2 (J, U). Since, for t ∈ J, also the weak convergence u j u f,q in L 1 ([0, t], U) is satisfied, with a similar reasoning as before we have What is left to show is that f ∈ P F (q). By (F3), (F4w) and the def- and lim inf c→∞ λ(c) Take the Hilbert space E = U := L 2 (K; R). Problem (40) can then be written, in abstract setting, in the form where r(t) := z(t, ·), w(t) := v(t, ·), y 0 := z 0 (·), y 1 := z 1 (·). The functions F : J × E → E and B : E → E are defined by The problem is well-posed by assumptions (a)-(e). Let A : D(A) = {r ∈ W 2,2 (K; R) : r(0) = r(L) = 0} → L 2 (K; R) be the Laplace operator Ar = r .

Condition (A) is satisfied and
It is easy to show that B : E × E → E is linear and bounded and then (B) is satisfied with B = b ∞ . Since the equation in (42) is single-valued, also (F1) is trivially satisfied. Now we prove (F2), by means of Theorem 2.3. Fix y = (y 1 , y 2 ) ∈ E and let e : E → R be linear and bounded. Hence there is ψ ∈ L 2 (K, R) satisfying e • F(t, y) = e (0, F (t, y 1 )) = L 0 ψ(x)ϕ y (t, x) dx, t ∈ J, with ϕ y as in (43). It is clear that L 0 h(·, ξ)y 1 (ξ)dξ is a Borel-measurable function in K, by (d) and the properties of y 1 . Hence, by (a)-(c), also the map (t, x) −→ ψ(x)ϕ y (t, x) is Borel-measurable in J × K. It implies that e • F(·, y) is measurable in J and then, by Theorem 2.3, condition (F2) is satisfied.
By means of Theorem 3.2(w1) we arrive to the following result Theorem 6.1. Consider Problem (40), assume conditions (a)-(e) and let then Problem (40) is controllable.
As in the previous example, system (45) can be written in the abstract form (42), with F : J × E → E defined by , r(x)).