Optimal Regulation of Impulsive Fractional Di ff erential Equation with Delay and Application to Nonlinear Fractional Heat Equation

The purpose of this paper was considering the impulsive fractional differential system with time delay. We investigated the existence of solution corresponding to the regulator in the admissible regulator set describing by the compact semigroup on Banach space. The result was applied to nonlinear fractional heat equation.


Introduction
There are many paper concerned with fractional differential equation and impulsive fractional differential equations.Such as, in 2008, Gastao, S. F., Frederico has solved the optimization regulation problem in the meaning of Caputo and the fractional Neether's theorem.In 2010, Zhongli Wei with the party have studied on the initial value problems for fractional differential equations in the sense of Riemann-Liouville fractional derivative.These papers are motivation of this research.
For this article, we discuss the nonlinear-retarded functional impulsive fractional differential equations ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D α t x(t) = Ax(t) + f (t, x(t), x(t − l), . . ., x(t − ml)) + B(t)u(t), t ∈ I\D Δx(t i ) = J i (x(t i )), t i ∈ D x(0) = x 0 , x(t) = ϕ(t), t ∈ [−ml, 0) (1) for some positive integer m and positive real l where I = [0, T ] , D = {t 1 , t 2 , . . ., t n }, A is a generator of a semigroup {S (t)} t≥0 on Banach space (B-space) X with some conditions, Δx(t i ) = x(t + i ) − x(t i ) denote the incitement of state x at time t i with the magnitude of incitement J i , i = 1, 2, ..., n.Then, we study a minimizing regulation problem of the system with the objective functional P, that is, to search u 0 an element of a family of admissible regulators A ad which P(u 0 ) ≤ P(u), for all u ∈ A ad . (2) In this case, define the objective functional by P(u) = T 0 r(t, x(t), x t , u(t))dt + g(x(T )) for all u ∈ A ad with a given running function r and a given terminal function g and x is a solution of the system consistent with a regulator u ∈ A ad .
The extent of this work starts with background, some important descriptions and theorems for substantiating the main resultant of the paper, for example, definition of fractional derivative, some fractional integral inequalities and the generalized of the Ascoli-Arzela Theorem (AAT) are recommended.The investigation of existence of solution for the system is then presented in Subsection 3.1.Furthermore, the optimization regulation problem is solved and written in Subsection 3.2.In the last section (Section 4), we solve the minimization regulation problem of nonlinear fractional heat by using our main result.

Background
Let X and Y be any B-spaces.Throughout this article, the symbol, L(X, Y) represent the space of bounded linear operators from X to Y. Especially, L(X) = L(X, X) which norm || • || L(X) .The symbol, PC ([a, b], X) represent the for positive, which 0 < n − α < 1, D n denote the ordinary derivative of order integer n and Γ(•) denote the Gamma function.
Continue this process until time interval [t m , T ].Then there exists a subsequence {x m }, s.t.
Therefore G is RCP.

Impulsive Integral Inequalities
Throughout this section, the family of all functions map from + to such that their derivatives of order α exist on + − {t k } and left continuous at t k , k ∈ {1, 2, . . ., n} for 0 < t k < t k+1 is represented by PC α ( + , ).
Proof.Setting the right hand side equal to μ(t).Then we have, Since Applying Theorem 4, we obtain
A2) there is some constant c s.t.
for all (x 1 , . . ., x m+1 ) ∈ X m+1 .A3) Y is another separable reflexive B-space that regulator u take the value, that is, for all x 1 , x 2 ∈ X and t ∈ I.

Existence of Solution of Impulsive Fractional Differential System
Definition 6 For any With assumptions A1-A5, the existence of solution of system (1) is proved by using the LSFPT and the compactness of semigroup {S (t)} t≥0 .From the Definition 6, define the operator F by Let x ∈ PC([−ml, T ], X).By using the continuity of ||x(t)|| and ||x(t − ml)|| and assumption A2, there is some positive constant N s.t.
Therefore, the operator F is bounded.
Lemma 7 Assume that assumption A1-A5 hold.Then the bounded operator F is continuous.
Proof.Suppose that x n is a sequence in PC([−ml, T ], X) converging to x in PC([−ml, T ], X).Then there is N 0 > 0 and for every n Corollary 8 Assume that assumption A1-A5 hold.The operator F maps bounded set into itself.
Proof.The proof is followed from the proof of Lemma 7.For each r > 0, there is some δ > 0 s.t. for every Lemma 9 Suppose assumptions A1-A5 hold.Then the F is operator compact.
Proof.Given a bounded subset B of PC([−ml, T ], X).By Corollary 8, F(B) is bounded.Define By continuity of φ(t) and compactness of For t ∈ (0, t 1 ] the Equation ( 17) reduce to Furthermore, the continuity of ||x(t)|| and ||x(t − ml)|| on (0, t 1 ) for all m = 1, . . ., k imply that there is a positive N s.t.||x(t)||, ||x(t − ml)|| ≤ N. By assumption A2 and assumption A4, there exist a positive constant c such that Hence, there exist RCP sets arbitrary close to the set On the interval (t 1 , t 2 ], we define ).By the condition A3, we get J 1 (Q(t 1 )) is RCP and this implies Q(t + 1 ) is also RCP.Let x(t + 1 ) = x 1 .Then for t ∈ (t 1 , t 2 ], the Equation (17) reduce to Generally, given any Similarly, for t ∈ (t k , t k+1 ], the Equation ( 17) reduce to By repeating these process till the time interval which expanded, Next, we will show that Q is equicontinuous on (t k , t k+1 ), k = 0, 1, . . ., n.Since B is bounded and follow from the inequality (18), there is an L u > 0 such that Let h > 0 and for 0 < t < t + h < t 1 and for x ∈ B. Then Hence F is equicontinuous on (0, t 1 ).In general, for (t k , t k+1 ), k = 0, 1, 2, . . ., n, for Using the same idea, we can show that Q is equicontinuous on (t k , t k+1 ) where k = 0, 1, 2, . . ., n.Therefore, Theorem 3 implies that F(B) is a RCP subset of PC([−ml, T ], X).Further, F is a compact operator.
Lemma 10 The set By using assumptions A2, there are Using assumption A3 and assumption A4, we get, Using Theorem 5, there is Theorem 11 Assume that assumptions A1-A5 hold, then the system (1) has at least one PC−mild solution corresponding to a regulator u ∈ A ad on [−ml, T ].
Proof.Define the operator F as ( 16).Then by Lemma 7 and Lemma 9, we have Then, the LSFPT to implies that F has a fixed point in PC([−ml, T ], X).Therefore, system (1) has at least PC−mild solution corresponding to the regulator u ∈ A ad on PC([−ml, T ], X).

Existence of Optimal Regulation
Let A ad be the admissible regulator set.Note that the result in Section 3.1 implies that for each regulator u ∈ A ad , there exits a PC-mild solution x corresponding to the regulator u.
Let us consider the minimization problem (P) corresponding to system (1).Find a regulator u 0 ∈ A ad s.t.
P(u 0 , x 0 ) ≤ P(u, x) for all u ∈ A ad which P(u, x) = T 0 r(t, x(t), x(t − l), . . ., x(t − ml), u(t))dt + g(x(T )) and x is a mild solution of system (1) corresponding to a regulator u ∈ A ad , the order pair (u, x) is called the admissible pair.For convenience, P(u, x) is short written by P(u).
We solve the optimizing regulation problem in this paper under the following assumptions, label by A6-A10; ) is sequentially lower semicontinuous on X m+1 × Y for a.e. on I. A8) r(t, ξ, •) is convex on Y for all ξ ∈ X m+1 and for a.e.t ∈ I. A9) There exist constants a i , b > 0 for all i = 1, . . ., m + 1 and η ∈ L 1 (I, ) s.t.
Theorem 12 Suppose that assumption A1-A10 hold.Then the problem (P) corresponding to system (1) has at least one solution, that is, there is an admissible pair (u 0 , x 0 ) such that P(u 0 , x 0 ) ≤ P(u, x) for all u ∈ A ad .
Proof.If inf{P(u)|u ∈ A ad } = +∞, it is well done.Suppose that inf{P(u)|u ∈ A ad } = w < +∞.By assumption A9, there are constants a i , b > 0 for all i = 0, . . ., m and η ∈ L 1 (I, ) such that Since η is non-negative, we get for some σ > 0, for all u ∈ A ad .Hence w ≥ −σ > −∞.By definition of minimum, there exists a minimizing sequence {u n } of P, that is lim n→∞ P(u n ) = w and So, there exists N 0 > 0 such that for some w 1 > 0 and hence ||u n || q L q (I,Y) ≤ w 1 +w c .Thus u n is a bounded sequence containing in the reflexive Bspace L q (I, Y).Therefore, u n has a convergence subsequence, relabelled as u n and u n → u 0 for some u 0 ∈ A ad = L q (I, Y).Let x n ∈ PC([−ml, T ], X) be a sequence of PC-mild solutions of system (1) corresponding to the regulators sequence The a priori estimate implies there is some positive constant ρ s.t.
We use the fact that x n (s) − x 0 (s) = 0 for s ∈ [−r, 0], so we have Applying Theorem 5, there exist M 1 > 0 independent on u, n and t such that The strongly continuous of ) for all t ∈ I. Then by assumption A6 and assumption A9, {r n (t)} is a sequence of non-negative measurable functions.So, by applying Fatou's Lemma, ≥ r(t, x 0 (t), x 0 t , u 0 (t))dt + g(x 0 (T )) = P(u 0 ).Therefore, P(u 0 ) = m.

Optimal Regulation of Nonlinear Fractional Heat Equation
Let us consider the boundary value problem with delay and regulation; ∂ α y(x, t) ∂t α = Δy(x, t) + f (x, t, y(x, t), y(x, t − l), . . ., y(x, t − ml)) where In this system, y(x, t) represents the temperature at the point x ∈ Ω at time t, the condition C3 represents the temperature at the histories time t ∈ [−ml, 0].The condition C4 represents the temperature on the boundary ∂Ω that is equal to zero.The input function f means an external heat sources.In this case, the function f depend on the histories data y(x, t−l), . . ., y(x, t−ml), which is impacted from the initial delay function ϕ(x, t) for t ∈ [−ml, 0] in the condition C3.Furthermore, the system is regulated by the regulator u with the sensor mapping Ω B(x, ξ)u(ξ, t)dξ.Given an admissible regulator set A ad = L q (Ω × I).We solve the optimization regulation problem (P 0 ) with the objective functional; where g ∈ C( , + ).That is, find u 0 ∈ A ad that minimize the objective functional.Let X = L p (Ω).For t ∈ [−ml, T ], define y(t): Ω → X by y(t)(x) = y(x, t) for all x ∈ Ω, and D α t y(t)x = ∂ α y(x, t) ∂t α , for all y ∈ X, x ∈ Ω.
We define Using the standard definition of the inner product, we define that for any y 1 , y 2 ∈ D(A); Therefore, A is symmetric and its eigenvalues must be real.Furthermore, for any y ∈ D(A), we have The right hand side vanishes only if y is constant but the only constant in D(A) is the zero constant.Thus, we obtain λ||y|| 2 = λy, y = Ay, y > 0, for all y 0 in D(A).

(C5)
This is precisely the definition of a positive operator, A is actually strongly positive.Because of Equation C5, the eigenvalues of A must be positive and we obtain a following lemma.

Lemma 13
The operator A is a infinitesimal generator of a compact C 0 -semigroup on X.

(C6)
Theorem 14 Suppose that the assumptions (HHf) and (HHJ) are sattisfied.Then the problem (P 0 ) for the nonlinear fractional heat equation with delay in N , C1 − C4 has at least one solution.
Proof.We solve the regulation problem (P 0 ) for system C1 − C4 pass the abstract form, C6.The definitions of f , J k (k = 1, 2, . . ., n) and the objective functional P and the assumptions (HHf), (HHJ) imply the assumptions of Theorem 11 and Theorem 12. Therefore, the problem (P 0 ) with system C1 − C4 has at least one solution.

Conclusions
In this work, we considered the fractional nonlinear differential system (1) with time lag, when A is the infinitesimal generator of a compact semigroup {S (t)} t≥0 satisfying the exponential stability.We proved the existence solution and solved the optimal regulation problem.We proposed a method for proving existence whose main component is the use of the Leray-Schauder Fixed Point Theorem(LSFPT).More precisely, we assume that the input function, f and operator J k , k ∈ {1, 2, . . ., n} satisfy the condition A1-A5.We successfully applied this method and use these assumptions to prove the existence of PC-mild solution.For studying the optimization regulation problem, we win to prove that system (1) has atleast one optimal regulator with conditions A1-A10.Beside the study of the solution and the optimization regulation problem, we give some examples (model of problem in the real world), we give example of f and J k , k ∈ {1, 2, . . ., n} such that satisfying the conditions (HHf) and (HHJ).Then we transform them to the abstract form and use our main results to conclude that these systems have atleast one optimal regulator.
Last but not least we should be interested in developing this method and use weakly assumptions to prove the existence and uniqueness of PC-mild solution a little further.Moreover, we should be interested in studying the others the solution behaviours for example;the stable property.Even though it seems likely that efforts in this direction can be successful, there no guarantee for that.Therefore, we can only hope for the best, but have to expect the worst.
x n (t), (x n ) t , u n (t))dt + g(x 0 (T )) Define an operator A: X → X as Ay = Δy for all y ∈ D(A) which D(A) contains all C 2 ( Ω) function vanishing on ∂Ω.Now we introduce the eigenvalue problem for the negative Laplacian; Ay = λy for all y ∈ D(A).