Fractional Integro-Di ff erential Equations of Mixed Type with Solution Operator and Optimal Controls

Received: March 2, 2011 Accepted: March 24, 2011 doi:10.5539/jmr.v3n3p140 The research is supported by the Centre of Excellence in Mathematics (CEM) of Thailand. Abstract Local and global existence and uniqueness of mild solution for the fractional integro-differential equations of mixed type with delay are proved by using a family of solution operators and the contraction mapping principle on Banach space. The Bolza optimal control problem of a corresponding controlled system is solved. The Gronwall lemma with singular and time lag is derived to be tool for obtaining a priori estimate. In addition, the application to the fractional nonlinear heat equation is shown.


Introduction
In this paper, we consider fractional integro-differential equations of mixed type with delay; on infinite dimensional Banach space X, where I = [0, T ], 0 < α ≤ 1, D α t denote the fractional derivative in the sense of Riemann-Liouville, f : I × X × X × X → X and ϕ ∈ C([−r, 0], X) are given, A is a linear operator corresponding to a solution operator {T α (t)} t≥0 in the Banach space X and G, S are nonlinear integral operators given by Gx(t) = t −r k(t, s)g(s, x(s))ds, S x(t) = T 0 h(t, s)q(s, x(s))ds. (2) Many research groups have studied and reported on integro-differential systems and fractional differential systems.These reports include the proof of the existence and uniqueness of a classical solution of an integro-differential equation by Chonwerayuth and a portion of work on the nonlinear impulsive integro-differential equations of mixed type by Wei.W. Furthermore, in 2009, Gisele M.Mophou proved existence and uniqueness of mild solution to impulsive fractional differential equations .
The scope of our work is to extend some results of these reports starting with preliminaries, some necessary definitions and theorems for proving main results such as description of fractional calculus and some different generalized Gronwall lemmas are introduced.The proof of the existence and uniqueness of solution for system (1) without control is then shown in Section 3.Moreover, the optimal control for system (1) via the Bolza cost functional is solved and reported in Section 4,.In the last section, we apply our result to fractional nonlinear heat equation.

Preliminaries
Let X be a Banach space and I = [0, T ], some important definitions and theorems those are used in this work are given as follows.
Definition 2.1.Let f : → X be a continuous(but not necessarily differentiable) function and let h > 0 denote a constant discretization span.The fractional difference of order α (α ∈ + ) of f is defined by the expression and its fractional derivative of order α is Definition 2.2.Assume that the function in the definition 2.1.has a Laplace 's transform.Then its fractional derivative of order α is defined by the following expression where 0 < α < 1, and the fractional integral of order α > 0 is defined by (5) These expression are called Riemann-Liouville definition, in particular, let f , u, v ∈ C( , X) and w be a real value function, we obtain some properties for 0 < α ≤ 1 see more detail in Jumarie G.
Let X and Y be two Banach spaces, L(X, Y) denote the space of bounded linear operators from whose moving norm is defined by From this moving norm, we generalize Gronwall lemma with time delay as follow.Using lemma 2.3, we devise the following new generalized Gronwall lemma which is very important for our work.
Lemma 2.4.Suppose x ∈ C([−r, T ], X) satisfies the following inequality The notion of solution operator plays a basic role in this study.We now consider a closed linear operator A densely defined in a Banach Space X and give a definition for the solution operator following.
Definition 2.5.Let A : X → X.For each α ∈ (0, 1], a family of bounded linear operators {T α (t)} t≥0 on X is called a solution operator corresponding to A if it satisfies the following conditions; 1. T α (t) is strongly continuous for t ≥ 0 and T α (0) = I; 3. Existence of Solutions to Fractional Integro-differential equations of mixed type Consider the nonlinear fractional system (1), where A : D(A) → X be an operator corresponding to a solution operator {T α (t)} t≥0 satisfying ||T α (t)|| L(X) ≤ Me ωt for some M ≥ 1, ω > 0 for all t ≥ 0, f : I × X × X × X → X and ϕ ∈ C([−r, T ], X) are given functions satisfies following conditions; (HF1) f : I × X × X × X → X is uniformly continuous in t and locally Lipschitz in x, ξ, η that for every τ > 0 and ρ > 0, there is a constant First of all, we study the properties of integral operators; We introduce the following assumptions (HG) and (HS ); (HG1) g : [−r, T ] × X → X is measurable in t on [−r, T ] and locally Lipschitz in x, i.e., let ρ > 0, there exists a constant (HG2) There exists a constant a g such that (HS1) q : I × X → X is measurable in t on I and locally Lipschitz in x, i.e., let ρ > 0, there exists a constant (HS2) There exists a constant a q and γ ∈ (0, 1) such that Using moving norm || • || B one can verify that integral operator G and S have the following properties.
Lemma 3.1.Under the assumption (HG), the operator G has the following properties; (1 (2) Given ρ > 0 and We can similarly obtain the following lemma.
Lemma 3.2.Under the assumption (HS ), the operator S has the following properties; (1) S : C(I, X) → C(I, X). ( Proof.The proof is similar to the proof of the lemma 3.1. Recall fractional integro-differential equations of mixed type (1), let 0 < α ≤ 1.By using ( 6) and ( 7), if x is a solution of (1), then the Since f is integrable, the right hand side of ( 12) is integrable in the sense of Bochner and apply w(0) = T α (t)ϕ(0) yields, then the system (1) is called mildly solvable on [−r, t 0 ] and this x is called a mild solution on [−r, t 0 ].
) is any solution of system (1) then x has an a priori bound, i.e., there we use (HF2), lemma 3.1 and lemma 3.2, there exists a constant L such that and By lemma 2.4, there exists a constant ρ > 0 such that ||x(t)|| ≤ ρ, for t ∈ I.
The existence and uniqueness of mild solution of ( 1) is then proved by constructed an operator F and proved that it is a strictly contraction by the following lemmas.
For each τ > 0, C τ ≡ C([−r, τ], X) with the usual supremum norm and for λ > 0, we set Then the map F is bounded.Indeed, by using ( 14), we obtain that Moreover, the properties of the map F are listed as following.
Theorem 3.6.Suppose (HF), (HS ), (HG) holds and A is a corresponding generator to a solution operator {T α (t)} t≥0 with exponentially bound.Then there exists a τ 0 > 0 such that the system (1) is mildly solvable on [−r, τ 0 ] and the mild solution is unique.
We break the main system (1) for a moment and consider the initial value problem, where A is an operator corresponding to the solution operator {T α (t)} t≥0 and f : [t 0 , T ] × X × X × X → X is continuous in t on [t 0 , T ] and uniformly Lipschitz continuous on X.We have the following results.
Definition 3.7.A continuous solution x of the integral equation, will be called a mild solution of the system (16).
Theorem 3.8.Under the assumptions (HF2), (HG) and (HS ), if f : [t 0 , T ] × X × X × X → X is continuous in t on [t 0 , T ] and uniformly Lipschitz continuous (with constant L) on X then for every x 0 ∈ X the system (16) has a unique mild solution x ∈ C([t 0 , T ], X).Moreover, the map Proof.For a given x 0 ∈ X, we define a mapping Then F is well-defined and bounded.For each x, y ∈ C([t 0 , T ], X), it follows readily from the definition of F, lemma 3.1 and lemma 3.2 that where M α is a bound of 1 αΓ(α) ||T α (t)|| on [t 0 , T ].Using ( 18), ( 19) and induction on n it follows that whence For n large enough (M α LT α ) n n!
< 1 and by a well-known extension of the contraction principle, F has a unique fixed point x in C([t 0 , T ], X).This fixed point is desired mild solution of ( 16).
The uniqueness of x and the Lipschitz condition of the map x 0 → x are consequences of the following argument.Let y be a mild solution of ( 16) on [t 0 , T ] with the initial value y 0 .Then, which yields both the uniqueness of x and the Lipschitz continuity of the map x 0 → x.
From the result of theorem 3.8, if f is uniformly Lipschitz, then we have the existence and uniqueness of a global mild solution for system (1).However, if we assume that f satisfies only local Lipschitz in x and uniformly continuous in t on bounded intervals, then we have the following local version of theorem 3.8.Theorem 3.9.Assume the assumptions of theorem 3.6 are holding.Then for every x 0 ∈ X, there is a t max ≤ ∞ such that the initial value problem has a unique mild solution x on [0, t max ).Moreover, if t max < ∞, then lim Proof.We start by showing that for every τ 0 ≥ 0 and x 0 ∈ X, and there exists a δ = δ(τ 0 , ||x 0 ||) such that the system ( 16) has a unique mild solution x on an interval [τ 0 , τ 0 + δ] whose length δ is define by, where L(c, t) is the local Lipschitz constant of f following from (HF1), lemma 3.1 and lemma 3.2, is given by ( 23).Define a map F by ( 18) maps the ball of radius ρ(τ 0 ) centered at 0 of C([τ 0 , τ 1 ], X) into itself as a result from the following estimation, where the last inequality is a consequence from the definition of τ 1 .In this ball, F satisfies a uniform Lipschitz condition with constant L = L(ρ(τ 0 ), τ 0 + 1) and thus in the proof of theorem 3.8, it possesses a unique fixed point x in the ball.This fixed point is the desired solution of ( 16) on the interval [τ 0 , τ 1 ].
Let [0, t max ) be the maximum interval of existence of mild solution x for ( 22).If t max < ∞, then lim if it is false, then there exists a sequence {t n } and C > 0 such that t n → t max and ||x(t n )|| ≤ C for all n, this implies that for each t n near enough to t max , x define on [0, t n ] can be extended to [0, t n + δ] where δ > 0 is independent of t n , hence x can be extend beyond t max , this contradicts the definition of t max .So if t max < ∞, then lim To prove the uniqueness of the local mild solution of ( 22) we note that if y is a mild solution of ( 22), then on every closed interval [0, τ 0 ] on which both x and y exist, they coincide by the uniqueness argument given in the end of the proof of theorem 3.8.Therefore, both x and y have the same t max and on [0, t max ), x = y.Theorem 3.10.If the assumptions of theorem 3.6 are holding, then the system (1) has a unique mild solution on [−r, T ].
Proof.Let [−r, t max ) be the maximum interval of existence of mild solution x for (1).If t max > T , there is nothing to prove.If t max < T , by theorem 3.9, then lim t→t max ||x(t)|| = +∞, contradicts with an a priori bound of solution.So the system (1) has a unique mild solution on [−r, T ].

Existence of Optimal Controls
In this section, the existence of optimal controls of system governed by the fractional integro-differential equation (1) will be discussed.
Suppose that A is a linear operator corresponding to a solution operator {T α (t)} t≥0 and Y is another separable reflexive Banach space from which the controls u take the values.Let U ad = L q (I, Y), 1 < q < ∞ denoting the admissible controls set.Consider the following controlled system; (HB) Suppose that B ∈ L(I, L(L q (I, Y), L p (I, X))) where 1 < q < ∞ and p > 1/α.Then B(•)u ∈ L p (I, X) for all u ∈ U ad and we give the definition of mild solution with respect to a control in U ad .
then x is said to be a mild solution with respect to (w.r.t.) u on [−r, T ].
Theorem 4.2.Under assumptions (HF), (HG), (HS ), (HB) and A is a linear operator corresponding to a solution operator {T α (t)} t≥0 with exponentially bound.Then for every u ∈ U ad , the system (24) has a unique mild solution w.r.t.u on [−r, T ].
Proof.Let u ∈ U ad , define f (t, x(t)) = f (t, x(t), Gx(t), S x(t)) + B(t)u(t) for all x ∈ X. Use the fact that B(•)u ∈ L p (I, X) for all u ∈ U ad and use assumption (HF), lemma 3.1 and lemma 3.2, we obtain that f satisfies the assumption (HF).By theorem 3.10, the system (24) has a unique mild solution w.r.t.u on [−r, T ].
We consider the Bolza problem (P 0 ): Find (x 0 , u 0 ) ∈ X × U ad such that J(x 0 , u 0 ) ≤ J(x u , u), for all u ∈ U ad (25) where J(x u , u) = T 0 l(t, x u (t), x u t , u(t))dt + Φ(x u (T )), for short, denoting by J(u) and x u denote the mild solution of the system (24) corresponding to the control u ∈ U ad .
We impose some assumptions for l, say (HL); is Borel measurable and Φ : X → is continuous and nonnegative.
3) l(t, x, y t , •) is convex on Y for each x, y t ∈ X and for a.e.t ∈ I.
4) There are a, b ≥ 0, c > 0 and η ∈ L 1 (I, ) such that l(t, x, y t , u) ≥ η(t) + a||x|| + b||y t || B + c||u|| q Y , for all t ∈ I and all x, y t ∈ X, u ∈ U ad A pair (x u , u) is said to be feasible if it satisfies equation ( 24).
Theorem 4.3.Suppose the assumption (HL) and the assumptions of theorem 4.2 hold.Then problem (P 0 ) admits at least one optimal pair.
for some ξ > 0, for all u ∈ U ad .Hence m ≥ −ξ > −∞.By definition of minimum, there exists a minimizing sequence {u n } of J , that is lim . This implies that u n is contained in a bounded subset of the reflexive Banach space L q (I, Y).So u n has a convergence subsequence relabeled as u n and u n → u 0 for some u 0 ∈ U ad = L q (I, Y).Let x n ⊆ C([−r, T ], X) be the corresponding sequence of solutions for the integral equation; From the a priori estimate, there exists a constant ρ > 0 such that ||x n || C([−r,T ],X) ≤ ρ for all n = 0, 1, 2, ... where x 0 denote the solution corresponding to u 0 , that is By (HF), (HG), (HS ), (HL), lemma 3.1 and lemma 3.2, for every t ∈ I there is a constant a(ρ) such that By using lemma 2.3, we found that ||x n (t) where M is a constant, is independent of u, n and t.Since B is strongly continuous, we have x n (T )) = T 0 l(t, x 0 (t), (x 0 n ) t , u 0 (t))dt + Φ(x 0 (T )) = J(u 0 ).

Application to Fractional Nonlinear Heat Equation
Consider the nonlinear heat equation control; = Δy(x, t) + f 1 (x, t, y(x, t)) + provided ||ξ||, || ξ|| ≤ ρ and s, t ∈ I.If we interpret y(x, t) as temperature at the point x ∈ Ω at time t, then the initial condition y(x, 0) means that the temperature at the initial time t = 0 is prescribed.Condition y(x, t) = 0, (x, t) ∈ ∂Ω × I means that the temperature on the boundary ∂Ω is equal to zero at any time.The function f describes an external heat sources.In this system, f and u are given.We then introduce the integral Gy(x, t) = t −r k(t − s)g(x, s, y(x, s))ds and S y(x, t) = T 0 h(t − s)q(x, s, y(x, s))ds, which directly impact to the system.Moreover, the system is controlled by controlling u via the sensor mapping Ω B(x, ξ)u(ξ, t)dξ.Let U ad = L q (Ω × I) be the admissible control set.We will solve the optimal problem (P 0 ) via the cost functional; where 0 < β ≤ 1, a ≥ 0, b(s) and c(s) are non-negative continuous functions.Then ||x(t)|| ≤ [||ϕ|| C + a]e bt β β , t ∈ I where b = sup s∈I [b(s) + c(s)].