# Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations

Authors: Xue Pan; Xiuwen Li; Jing Zhao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.


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## Body

## 1. Introduction

The purpose of this paper is to investigate the solvability and optimal controls for the following semilinear control systems with Riemann-Liouville fractional derivatives:(1)Dtαx(t)L=Ax(t)+Bu(t)+f(t,x(t)),dddddddddddddt∈(0,b],u∈Uad,I0+1-αx(t)|t=0=x0∈X,
where 0<α≤1, DtαL denotes the Riemann-Liouville fractional derivative of order α with the lower limit zero. A:D(A)⊆X→X is the infinitesimal generator of a C0-semigroup T(t)(t≥0) on a separable Banach space X. f:[0,b]×X→X is a given function to be specified later. The control function u is given in a suitable admissible control set Uad. B is a linear operator from a separable reflexive Banach space Y into X. The cost functional over the family of admissible state control pair (x,u) is given by
(2)𝒥(x,u)=∫Jℒ(t,x(t),u(t))dt.In recent years, fractional calculus has been paid more and more attention that lies on the fact that it allows us to consider integration and differentiation of any order, not necessarily integer, and fractional-order models are more accurate than integer-order models; that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of term which insures the history and its impact on the present and future in a model. Therefore, it has drawn great applications of the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, viscoelasticity, heat conduction, electricity mechanics, control theory, and so forth. For more details on these topics one can see, for instance, [1–17] and the reference therein.The definitions of Riemann-Liouville fractional derivatives or integrals initial conditions play an important role in some practical problems. Heymans and Podlubny [18] have demonstrated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals on the field of the viscoelasticity, and such initial conditions are more appropriate than physically interpretable initial conditions.Since the pioneering work on the fundamental solutions of Caputo fractional evolution equations associated with some probability densities has been reported by El-Borai [19, 20], the study of the existence, controllability, and optimal controls of the fractional semilinear functional differential equations and inclusions with the infinitesimal generator A of a C0-semigroup have been extensively studied (one can see [21–23]). However, to the best of our knowledge, the solvability and optimal controls for fractional semilinear differential equations with Riemann-Liouville fractional derivatives are still untreated topics in the literature and this fact is the motivation of the present work. Our aim in this paper is to provide some suitable sufficient conditions for the existence and uniqueness of solutions and optimal control results corresponding to the admissible control sets of fractional abstract Cauchy problems with the Riemann-Liouville fractional derivatives.The rest of this paper is organized as follows. In Section2, we will present some basic definitions and preliminary facts, such as definitions, lemmas, and theorems, which will be used throughout the following sections. In Section 3, by applying the well-known fixed point theorem, some sufficient conditions are established for the existence and uniqueness of mild solutions of the system (1). In Section 4, we will study the optimal controls for semilinear differential equations with Riemann-Liouville fractional derivatives. Finally, we present an example to demonstrate our main results in Section 5.


## 2. Preliminaries

In this section, we introduce some basic definitions and preliminaries which are used throughout this paper. The norm of a Banach spaceX will be denoted by ∥·∥X. Lb(X,Y) denotes the space of bounded linear operators from X to Y. For the uniformly bounded C0-semigroup T(t), we set M:=sup⁡t≥0∥T(t)∥Lb(X)<∞. Let C(J,X) denote the Banach space of all X-value continuous functions from J=[0,b] into X with the norm ∥x∥C=sup⁡t∈J∥x(t)∥X. Let AC(J,X) be the space of functions f which are absolutely continuous on J and ACm(J,X)={f:J→X and f(m-1)(x)∈AC(J,X)}. To define the mild solutions of (1), we consider the Banach space C1-α(J,X)={x:t1-αx(t)∈C(J,X)} with the norm ∥x∥C1-α=sup⁡{t1-α∥x(t)∥X:t∈J}. Obviously, the space C1-α(J,X) is a Banach space.Firstly, let us recall the following definitions from fractional calculus. For more details, one can see [3, 16].Definition 1.
The integral(3)Itαf(t)=1Γ(α)∫0t(t-s)α-1f(s)ds,α>0,
is called Riemann-Liouville fractional integral of order α, where Γ is the gamma function.Definition 2.
For a functionf(t) given in the interval [0,∞), the expression
(4)LDtαf(t)=1Γ(n-α)(ddt)n∫0t‍(t-s)n-α-1f(s)dt,
where n=[α]+1, [α] denotes the integer part of number α and is called the Riemann-Liouville fractional derivative of order α.Lemma 3 (see [3]).
Letα>0, let m=[α]+1, and let xm-α(t)=I0+m-αx(t) be the fractional integral of order m-α. If x(t)∈L1(J,X) and xm-α(t)∈ACm(J,X), then one has the following equality:
(5)ItαDtαx(t)=Lx(t)-∑k=1mxm-α(m-k)(0)Γ(α-k+1)tα-k.The Laplace transform formula for the Riemann-Liouville fractional integral is defined by(6)L{Itαu(t);λ}=1λαu^(λ),
where u^(λ) is the Laplace of u defined by
(7)u^(λ)=∫0∞‍e-λtu(t)dt,Reλ>ω,|u(t)|≤ceωt,cisaconstant.Lemma 4.
Letα∈(0,1] and h∈Lp(J,X), p>1/α, if x(t)∈L1(J,X), x1-α(t)∈AC(J,X), and x is a solution of the following problem:
(8)Dtαx(t)L=Ax(t)+h(t),t∈(0,b],I0+1-αx(t)|t=0=x0∈X;
then, x satisfies the following equation:
(9)x(t)=tα-1Tα(t)x0+∫0t(t-s)α-1Tα(t-s)h(s)ds,t∈J,
where
(10)Tα(t)=α∫0∞θξα(θ)T(tαθ)dθ,ξα(θ)=1αθ-1-(1/α)ϖα(θ-(1/α)),ϖα(θ)=1π∑n=1∞(-1)n-1θ-nα-1Γ(nα+1)n!sin(nπα),fffffffffffffffffffffffθ∈(0,∞);
and ξα is a probability density function defined on (0,∞), that is,
(11)ξα(θ)≥0,θ∈(0,∞),∫0∞‍ξα(θ)dθ=1.Proof.
Apply Riemann-Liouville fractional integral operator on both sides of (8); then, by Lemma 3, we get
(12)x(t)=I0+1-αx(t)|t=0Γ(α)tα-1+ItαAx(t)+Itαh(t)=tα-1Γ(α)x0+ItαAx(t)+Itαh(t);
that is,
(13)x(t)=tα-1Γ(α)x0+1Γ(α)∫0t‍(t-s)α-1[Ax(s)+h(s)]ds.
Let λ>0; taking the Laplace transformations
(14)x^(λ)=∫0∞‍e-λtx(t)dt,h^(λ)=∫0∞‍e-λth(t)dt,
to (13), we obtain
(15)x^(λ)=1λαx0+1λαAx^(λ)+1λαh^(λ)=(λαI-A)-1x0+(λαI-A)-1h^(λ)=∫0∞e-λαtT(t)x0dt+∫0∞e-λαtT(t)h^(λ)dt.
Consider the one-sided stable probability density(16)ϖα(θ)=1π∑n=1∞‍(-1)n-1θ-nα-1Γ(nα+1)n!sin⁡(nπα),ddddddddddddddddddddddddddθ∈(0,∞),
whose Laplace transform is given by
(17)∫0∞e-λθϖα(θ)dθ=e-λα,α∈(0,1).
Hence, it follows from (15) and (17) that
(18)∫0∞e-λαtT(t)x0dt=∫0∞e-(λs)αT(sα)x0dsα(t=sα)=α∫0∞e-(λs)αT(sα)sα-1x0ds=α∬0∞ϖα(θ)e-λsθT(sα)sα-1x0dθds=α∬0∞ϖα(θ)e-λuT(uαθα)sα-1θαx0dudθ(u=θs)=∫0∞e-λu[α∫0∞ϖα(θ)T(uαθα)sα-1θαx0dθ]du,∫0∞e-λαtT(t)h^(λ)dt=∫0∞e-λαtT(t)[∫0∞e-λsh(s)ds]dt=∬0∞e-λαtT(t)e-λsh(s)dsdt=∬0∞e-(λμ)αT(μα)qμα-1e-λsh(s)dsdμggggggggggggggggddddddd(t=μα)=∭0∞αϖα(θ)e-λμθT(μα)μα-1e-λsh(s)dθdsdμ=∭0∞αϖα(θ)e-λνT(ναθα)να-1θαe-λsh(s)dνdθdsffffffffffffffffffffffffdddddddd(ν=μθ)=∭0∞αϖα(θ)e-λ(ν+s)T(ναθα)να-1θαh(s)dνdθds=∫0∞‍∫0∞‍∫s∞‍αϖα(θ)e-λτT((τ-s)αθα)dddddddddd×(τ-s)α-1θαh(s)dτdθds(τ=ν+s)=∫0∞e-λτdddd×[(τ-s)α-1θα∫0∞e-λτα∫0τ‍∫0∞‍ϖα(θ)T((τ-s)αθα)dddddddddddd×(τ-s)α-1θαh(s)dθds]dτ.
According to the above work, we get(19)x^(λ)=∫0∞e-λt[α∫0∞ϖα(θ)T(tαθα)tα-1θαx0dθfffffffffffffddddddddddddd+α∫0t∫0∞ϖα(θ)T((t-s)αθα)fffffffffddddddddddddddddddddd×(t-s)α-1θαh(s)dθds]dt.
Now, we can invert the Laplace transform to (19) and obtain
(20)x(t)=α∫0∞‍ϖα(θ)T(tαθα)tα-1θαx0dθffffff+α∫0t‍∫0∞‍ϖα(θ)T((t-s)αθα)(t-s)α-1θαh(s)dθdt=α∫0∞‍1αθ-1-(1/α)ϖα(θ-(1/α))θT(tαθ)tα-1x0dθddddd+α∫0t‍∫0∞‍(t-s)α-11αθ-1-(1/α)ϖαddddddddddd×(θ-(1/α))θT((t-s)αθ)h(s)dθdt.
Let(21)ξα(θ)=1αθ-1-(1/α)ϖα(θ-(1/α)),Tα(t)=α∫0∞θξα(θ)T(tαθ)dθ.
Then, we get(22)x(t)=tα-1Tα(t)x0+∫0t(t-s)α-1Tα(t-s)h(s)ds.
This completes the proof of the lemma.According to Lemma4, we give the following definition.Definition 5.
A functionx∈C1-α(J,X) is called a mild solution of (1) if it satisfies the following fractional integral equation:
(23)x(t)=tα-1Tα(t)x0+∫0t(t-s)α-1Tα(t-s)Bu(s)ds+∫0t(t-s)α-1Tα(t-s)f(s,x(s))ds.Remark 6.
A mild solutionx(·)∈C1-α(J,X) of the system (1) is referred to as a state trajectory of the fractional semilinear differential equation corresponding to the initial state x0 and the control u(·).Due to the paper [23], we can obtain the following.Lemma 7.
The operatorTα(t) has the following properties.(i)
For any fixedt≥0,  Tα(t) is linear and bounded operators; that is, for any x∈X,
(24)∥Tα(t)x∥≤αMΓ(1+α)∥x∥.(ii)
Tα(t)(t≥0) is strongly continuous.(iii)
For anyt>0,  Tα(t) is also a compact operator if T(t) is compact.Let us recall the following generalized Gronwall inequality which can be found in [24].Lemma 8.
Supposeβ>0, a(t) is a nonnegative function locally integrable on [0,η], and b(t) is a nonnegative, nondecreasing continuous function defined on [0,η] and b(t)≤M (constant), and suppose y(t) is nonnegative and locally integrable on [0,η] with
(25)y(t)≤a(t)+b(t)∫0t(t-s)β-1y(s)ds,t∈[0,η].
Then,
(26)y(t)≤a(t)+∫0t‍[∑n=1∞‍[b(t)Γ(β)]nΓ(nβ)(t-s)nβ-1a(s)]ds,dddddddddddddddddddddddddddddddt∈[0,η].Remark 9.
Under the hypotheses of Lemma8, let a(t) be a nondecreasing function on [0,η]. Then,
(27)y(t)≤a(t)Eβ(b(t)Γ(β)tβ),
where Eβ is the Mittag-Leffler function defined by
(28)Eβ(z)=∑k=0∞‍zkΓ(kβ+1).


## 3. Existence of Mild Solutions

This section is devoted to the study of the existence and uniqueness results for a class of semilinear differential equations with Riemann-Liouville fractional derivatives.In what follows, we will make the following hypotheses on the data of our problems:H(1):
the functionf:J×X→X satisfies the following:
(i)
f(·,x):J→X is measurable for all x∈X and f(t,·):X→X is continuous for a.e. t∈J;(ii)
there exists a functionϕ(·)∈Lp(J,R+), p>1/α, and a constant ρ>0 such that
(29)∥f(t,x(t))∥≤ϕ(t)+ρt1-α∥x(t)∥,fora.e.t∈J;(iii)
there exists a functionφ(·)∈Lp(J,R+), p>1/α, such that
(30)∥f(t,x)-f(t,y)∥≤φ(t)∥x-y∥C1-α,fora.e.t∈Jandallx,y∈C1-α(J,X);H(2):
the operatorB∈Lb(Lp(J,Y),Lp(J,X));H(3):
the multivalued mapU:J→Pf(Y) (where Pf(Y) is a class of nonempty closed and convex subsets of Y) is measurable and there exists a function u(t)∈Lp(J,R+), p>1/α, such that
(31)∥U(t)∥=sup⁡{∥v∥:v∈U(t)}≤u(t),fora.e.t∈J.Set the admissible control set(32)Uad=SUp={u∈Lp(J):u(t)∈U(t)a.e.},p>1α.Then,Uad≠∅ (see Proposition 2.1.7 and Lemma 2.3.2 of [25]). And it is not difficult to check that Uad is a closed and convex subset of Lp(J,Y).In order to discuss the solvability and optimal control of system (1), we need to consider the following.Lemma 10.
Assume thatH(1)–H(3) hold. Then, there exists a constant ω>0 such that
(33)∥x∥C1-α≤ω,foranysolutionxof(1).Proof.
Ifx is a mild solution of system (1) with respect to u∈Uad on C1-α(J,X), then
(34)x(t)=tα-1Tα(t)x0+∫0t(t-s)α-1Tα(t-s)Bu(s)ds+∫0t(t-s)α-1Tα(t-s)f(s,x(s))ds.
Fort∈J, we obtain that
(35)t1-α∥x(t)∥≤∥Tα(t)x0∥+t1-α∫0t(t-s)α-1∥Tα(t-s)Bu(s)∥ds+t1-α∫0t(t-s)α-1∥Tα(t-s)f(s,x(s))∥ds≤αMΓ(1+α)dd×[∥x0∥+t1-α∫0t(t-s)α-1∥(Bu)(s)∥dsddd+t1-α∫0t(t-s)α-1‍ddddddddddd×[ϕ(s)+ρs1-α∥x(s)∥]dst1-α∫0t(t-s)α-1∥(Bu)(s)∥dst1-α∫0t(t-s)α-1∥(Bu)(s)∥ds∫∫]≤αMΓ(1+α)dd×[∥x0∥+(p-1pα-1)(p-1)/pb1-(1/p)dddd×(∥Bu∥Lp+∥ϕ∥Lp)ddd+ρb1-α∫0t‍(t-s)α-1s1-α∥x(s)∥ds(p-1pα-1)(p-1)/p∫].
Let(36)κ=αMΓ(1+α)[∥x0∥+(p-1pα-1)(p-1)/pb1-(1/p)dddddddddd×(∥Bu∥Lp+∥ϕ∥Lp)(p-1pα-1)(p-1)/p∫∫],W(t)=t1-α∥x(t)∥;
then by (35), we have
(37)W(t)≤κ+αMρb1-αΓ(1+α)∫0t(t-s)α-1W(s)ds.
It follows from Remark9 that
(38)W(t)≤κEα(Mρb):=ω.
Therefore,∥x∥C1-α=sup⁡t∈Jt1-α∥x(t)∥≤ω. The proof is completed.Now, we are in the position to present our first result.Theorem 11.
Assume that the hypothesesH(1)–H(3) are satisfied. Then, the problem (1) has a unique mild solution on J provided that
(39)αMΓ(1+α)(p-1pα-1)1-(1/p)b1-(1/p)∥φ∥Lp<1,MbαΓ(1+α)<1.Proof.
Consider the operatorϜ:C1-α(J,X)→C1-α(J,X) defined by
(40)(Ϝx)(t)=tα-1Tα(t)x0+∫0t(t-s)α-1Tα(t-s)Bu(s)ds+∫0t(t-s)α-1Tα(t-s)f(s,x(s))ds.
Choose(41)r≥(αMΓ(1+α)[∥x0∥+(p-1pα-1)(p-1)/pfffddddddddd×bα-(1/p)(∥Bu∥Lp+∥ϕ∥Lp)(p-1pα-1)(p-1)/p∫∫])ddd×(1-MbαΓ(1+α))-1,
and let Br={x∈C1-α(J,X):∥x∥≤r}. It is obvious that Br is a bounded, closed, and convex subset of C1-α(J,X).
Firstly, we show thatϜ maps Br into itself. In fact, for any x∈Br and t∈J, like the proof of Lemma 10, we can easily obtain ∥(Ϝx)(t)∥≤r which means that ϜBr⊆Br.
Next, we show thatϜ is a contraction operator on C1-α(J,X).
Indeed, lett∈J, and x,y∈C1-α(J,X); then we obtain
(42)t1-α∥(Ϝx)(t)-(Ϝy)(t)∥≤t1-α∫0t(t-s)α-1dddddd×∥Tα(t-s)[f(s,x(s))-f(s,y(s))]∥ds≤αMΓ(1+α)t1-α∫0t(t-s)α-1φ(s)ddddddddddddd×∥x-y∥C1-αds≤αMΓ(1+α)(p-1pα-1)1-(1/p)b1-(1/p)dd×∥φ∥Lp∥x-y∥C1-α.
Since(αM/(Γ(1+α)))((p-1)/(pα-1))1-(1/p)b1-(1/p)∥φ∥Lp<1, so Ϝ is a contradiction operator. According to Banach’s fixed point theorem, we obtain the problem (1) that has a unique mild solution on J. The proof is completed.Remark 12.
In Theorem11, we investigate the existence of a local mild solution of (1). Next, if we assume that the semigroup T(t)(t>0) is compact, then we can get the global version of a mild solution for system (1).Firstly, for a compact semigroup, we have the following.Lemma 13 (see [26]).
LetT(t) be a C0-semigroup. If T(t) is a compact semigroup, then T(t) is continuous in the uniform operator topology for t>0.The key tool in the existence result is the following Schaefer fixed point theorem.Theorem 14.
LetX be a Banach space and let F:X→X be a completely continuous operator. If the set
(43)Ω(F)={x∈X:x=λF(x),λ∈(0,1)}
is bounded, then F has at least a fixed point.Now, we are ready to state the existence result which is based on Theorem14.Theorem 15.
Assume thatH(1)–H(3) hold; if A is the infinitesimal generator of a compact semigroup T(t), then the problem (1) has at least one mild solution on J.Proof.
We consider the operatorϜ:C1-α(J,X)→C1-α(J,X) defined by
(44)(Ϝx)(t)=tα-1Tα(t)x0+∫0t(t-s)α-1Tα(t-s)Bu(s)ds+∫0t(t-s)α-1Tα(t-s)f(s,x(s))ds.
For the sake of convenience, we subdivide the proof into several steps.
Step
1.
Ϝ maps bounded sets into bounded sets in C1-α(J,X).
In fact, it is enough to show that, for anyr>0, there exists a η>0 such that, for each x∈Br={x∈C1-α(J,X):∥x∥C1-α≤r}, we have ∥Ϝx∥C1-α≤η.
For eacht∈J, we obtain
(45)t1-α∥(Ϝx)(t)∥≤∥Tα(t)x0∥+t1-α∫0t(t-s)α-1∥Tα(t-s)(Bu)(s)∥ds+t1-α∫0t(t-s)α-1∥Tα(t-s)f(s,x(s))∥ds≤αMΓ(1+α)dd×[∥x0∥+t1-α∫0t(t-s)α-1∥(Bu)(s)∥dsddd+t1-α∫0t(t-s)α-1dddddddddd×[ϕ(s)+ρs1-α∥x(s)∥]ds∫0t(t-s)α-1∥(Bu)(s)∥ds∫∫]≤αMΓ(1+α)dd×[∥x0∥+(p-1pα-1)(p-1)/pb1-(1/p)dddddd×(∥Bu∥Lp+∥ϕ∥Lp)(p-1pα-1)(p-1)/p∫∫]+ρMbrΓ(1+α),
which implies that
(46)∥Ϝx∥C1-α≤αMΓ(1+α)dd×[∥x0∥+(p-1pα-1)(p-1)/pddddd×b1-(1/p)(∥Bu∥Lp+∥ϕ∥Lp)(p-1pα-1)(p-1)/p(p-1pα-1)(p-1)/p∫]+ρMbrΓ(1+α):=η.
Thus, we know that{t1-α(Ϝx)(t):x∈Br} is a bounded set in C(J,X).
Step
2. We prove that Ϝ is continuous.
Let{xn} be a sequence such that xn→x in C1-α(J,X) as n→∞. Then, for each t∈J, we obtain
(47)t1-α∥(Ϝxn)(t)-(Ϝx)(t)∥≤t1-α∫0t‍(t-s)α-1∥Tα(t-s)dddddddddddddddd×[f(s,xn(s))-f(s,x(s))]∥ds≤αMΓ(1+α)t1-αddd×∫0t‍(t-s)α-1φ(s)∥xn-x∥C1-αds≤αMΓ(1+α)ddd×(p-1pα-1)1-(1/p)b1-(1/p)∥φ∥Lp∥xn-x∥C1-α.
Hence, we get(48)∥Ϝxn-Ϝx∥C1-α≤αMΓ(1+α)dd×(p-1pα-1)1-(1/p)b1-(1/p)dd×∥φ∥Lp∥xn-x∥C1-α⟶0asn⟶∞.
Step
3. We prove that {t1-α(Ϝx)(t):x∈Br} is an equicontinuous set in C(J,X).
Firstly, for anyε>0, by Lemma 7(ii), Tα(t) is strongly continuous; then, there exists a δ′>0, such that
(49)∥Tα(τ2)x0-Tα(τ1)x0∥<ε2,as|τ2-τ1|<δ′.
Thus, for the aboveε>0, there exists δ0=min⁡{δ′,((pα-1)/(p-1))[Γ(α)ε/(2M(∥Bu∥Lp+∥ϕ∥Lp+ρr))]p/(p-1)}>0, such that, for any x∈Br, τ1=0, 0<τ2≤δ0, one can obtain
(50)∥τ21-α(Ϝx)(τ2)-τ11-α(Ϝx)(τ1)∥≤∥Tα(τ2)x0-x0∥+∥τ21-α∫0τ2‍(τ2-s)α-1Tα(τ2-s)Bu(s)ds∥+∥τ21-α∫0τ2‍(τ2-s)α-1Tα(τ2-s)f(s,x(s))ds∥≤∥Tα(τ2)x0-x0∥+αMΓ(1+α)(p-1pα-1)(p-1)/p×(∥Bu∥Lp+∥ϕ∥Lp+ρr)δ01-(1/p)<ε2+ε2=ε.
Hence, by the definition of equicontinuity, we get thatϜ is equicontinuous on [0,δ0].
Next, for anyx∈Br and (δ0/2)≤τ1<τ2≤b, we obtain
(51)∥τ21-α(Ϝx)(τ2)-τ11-α(Ϝx)(τ1)∥≤∥Tα(τ2)x0-Tα(τ1)x0∥+∥∫τ21-α∫0τ2‍(τ2-s)α-1Tα(τ2-s)Bu(s)dsddd-t1-α∫0τ1‍(τ1-s)α-1Tα(τ1-s)Bu(s)ds∥+∥τ21-α∫0τ2‍(τ2-s)α-1ddddddddd×Tα(τ2-s)f(s,x(s))ds-τ11-αdddd×∫0τ1‍(τ1-s)α-1Tα(τ1-s)f(s,x(s))ds∥≤∥Tα(τ2)x0-Tα(τ1)x0∥+[τ21-α-τ11-α]∥∫0τ1(τ1-s)α-1Tα(τ1-s)Bu(s)ds∥+τ21-α∥∫0τ1[(τ1-s)α-1-(τ2-s)α-1]ddddddddd×Tα(τ1-s)Bu(s)ds∫0τ1[(τ1-s)α-1-(τ2-s)α-1]∥+τ21-α∥∫0τ1(τ1-s)α-1[Tα(τ2-s)dddddddddddddddddd-Tα(τ1-s)]Bu(s)ds∫0τ1[(τ1-s)α-1-(τ2-s)α-1]∥+τ21-α∥∫τ1τ2(τ2-s)α-1Tα(τ2-s)Bu(s)ds∥+[τ21-α-τ11-α]∥∫0τ1(τ1-s)α-1dddddddddddddddddd×Tα(τ1-s)f(s,x(s))ds∫0τ1[(τ1-s)α-1-(τ2-s)α-1]∥+τ21-α∥∫0τ1[(τ1-s)α-1-(τ2-s)α-1]dddddddddd×Tα(τ1-s)f(s,x(s))ds∫0τ1[(τ1-s)α-1-(τ2-s)α-1]∥+τ21-α∥∫0τ1(τ1-s)α-1[Tα(τ2-s)ddddddddddddddddddd-Tα(τ1-s)]f(s,x(s))ds∫0τ1[(τ1-s)α-1-(τ2-s)α-1]∥+τ21-α∥∫τ1τ2(τ2-s)α-1Tα(τ2-s)f(s,x(s))ds∥≤Q1+Q2+Q3+Q4+Q5+Q6+Q7+Q8+Q9.
By the assumptions and Holder’s inequality, we have(52)Q2≤[τ21-α-τ11-α]αMΓ(1+α)×∫0τ1(τ1-s)α-1∥Bu(s)∥ds≤αMΓ(1+α)(p-1pα-1)(p-1)/pddd×∥Bu∥Lpτ1α-(1/p)[τ21-α-τ11-α].
Similarly, we obtain(53)Q3≤2αMτ21-αΓ(1+α)(p-1pα-1)(p-1)/pddd×∥Bu∥Lp(τ2-τ1)α-(1/p),Q5≤αMτ21-αΓ(1+α)(p-1pα-1)(p-1)/pddd×∥Bu∥Lp(τ2-τ1)α-(1/p),Q6≤αMΓ(1+α)(p-1pα-1)1-(1/p)××[∥ϕ∥Lp+ρω]τ1α-(1/p)[τ21-α-τ11-α],Q7≤2αMτ21-αΓ(1+α)(p-1pα-1)1-(1/p)ddd×[∥ϕ∥Lp+ρω](τ2-τ1)α-(1/p),Q9≤αMτ21-αΓ(1+α)(p-1pα-1)1-(1/p)dd×[∥ϕ∥Lp+ρω](τ2-τ1)α-(1/p),
and, for Q4, Q8, there exists a δ>0 which is small enough, such that
(54)Q4≤τ21-α[∥∫0τ1-δ(τ1-s)α-1[Tα(τ2-s)dddddddddddddddddddd-Tα(τ1-s)]Bu(s)ds∫0τ1-δ(τ1-s)α-1[Tα(τ2-s)∫∫∫∥ddddddd+∥∫τ1-δτ1(τ1-s)α-1[Tα(τ2-s)dddddddddddddddddddddd-Tα(τ1-s)]Bu(s)ds∫τ1-δτ1(τ1-s)α-1[Tα(τ2-s)∫∫∫∥∥∫0τ1-δ(τ1-s)α-1[Tα(τ2-s)∫τ1-δτ1(τ1-s)α-1[Tα(τ2-s)∫τ1-δτ1(τ1-s)α-1[Tα(τ2-s)∫]≤sup⁡s∈[0,τ1-δ]⁡∥Tα(τ2-s)dddddddddddd-Tα(τ1-s)∥τ21-α(p-1pα-1)(p-1)/p×∥Bu∥Lp(τ1(αp-1)/(p-1)-δ(αp-1)/(p-1))1-(1/p)+2αMτ21-αΓ(1+α)(p-1pα-1)(p-1)/p∥Bu∥Lpb1-αδ1-(1/p),Q8≤sup⁡s∈[0,τ1-δ]∥Tα(τ2-s)-Tα(τ1-s)∥×τ21-α(p-1pα-1)(p-1)/p[∥ϕ∥Lp+ρω]×(τ1(αp-1)/(p-1)-δ(αp-1)/(p-1))1-(1/p)+2αMτ21-αΓ(1+α)(p-1pα-1)(p-1)/p[∥ϕ∥Lp+ρω]b1-αδ1-(1/p).
Since the compactness ofT(t)(t>0) and Lemma 13 imply the continuity of Tα(t)(t>0) in t in the uniform operator topology, it can be easily seen that Q4 and Q8 tend to zero independently of x∈Br as τ2→τ1, δ→0. It is also clear that Qi(i=1,2,3,5,6,7,9) tend to zero as τ2→τ1 does not depend on particular choice of x. Thus, we get that ∥τ21-α(Ϝx)(τ2)-τ11-α(Ϝx)(τ1)∥ tends to zero independently of x∈Br as τ2→τ1 which implies that Ϝ is equicontinuous on δ0/2≤τ1<τ2≤b.
Therefore, by all of the above work, we can get that{t1-α(Ϝx)(t):x∈Br} is an equicontinuous set in C(J,X).
Step
4. We show that Ϝ is compact.
Lett∈J be fixed; we show that the set Π(t)={t1-α(Ϝx)(t):x∈Br} is relatively compact in X.
Clearly,Π(0)={x0} is compact, so it is only necessary to consider t>0. For each ϵ∈(0,t), t∈(0,b], and x∈Br and any δ>0, we define
(55)Πϵ,δ(t)={t1-αϜϵ,δ(x)(t):x∈Br},
where
(56)t1-αϜϵ,δ(x)(t)=Tα(t)x0+αt1-α×∫0t-ϵ‍∫δ∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ)dddddddd×[Bu(s)+f(s,x(s))]dθds.=Tα(t)x0+αT(ϵαδ)t1-α×∫0t-ϵ‍∫δ∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ-ϵαδ)dddddddd×[Bu(s)+f(s,x(s))]dθds.
From the boundedness of∫0t-ϵ‍∫δ∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ-ϵαδ)[Bu(s)+f(s,x(s))]dθds and the compactness of T(ϵαδ)(ϵαδ>0), we obtain that the set Πϵ,δ(t)={t1-αϜϵ,δ(x)(t):x∈Br} is relatively compact set in X for each ϵ∈(0,t) and δ>0. Moreover, we have
(57)t1-α∥Ϝ(x)(t)-Ϝϵ,δ(x)(t)∥=t1-α∥α∫0t‍∫0∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ)dddddddddd×[Bu(s)+f(s,x(s))]dθdsdddd-α∫0t-ϵ‍∫δ∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ)dddddddddddd×[Bu(s)+f(s,x(s))]dθdsα∫0t‍∫∫∥≤αt1-α∥∫0t∫0δθ(t-s)α-1ξα(θ)T((t-s)αθ)dddddddddd×[Bu(s)+f(s,x(s))]dθds∫0t∫0δθ(t-s)α-1ξα(θ)T((t-s)αθ)∫0δ‍∫∫0δ‍∥+αt1-α∥∫t-ϵt∫δ∞‍‍θ(t-s)α-1ξα(θ)T((t-s)αθ)ddddddddddddd×[Bu(s)+f(s,x(s))]dθds∫t-ϵt∫δ∞‍‍∫∥≤αM(p-1pα-1)1-(1/p)[∥Bu∥Lp+∥ϕ∥Lp+rρ]×[b1-(1/p)∫0δ‍θξα(θ)dθ+b1-αΓ(1+α)ϵα-(1/p)].
Since∫0∞‍θξα(θ)dθ=1/(Γ(1+α)), the last inequality tends to zero when ϵ→0 and δ→0. Therefore, there are relatively compact sets arbitrarily close to the set Π(t)(t>0). Hence, the set Π(t)(t>0) is also relatively compact in X.
Step
5. A priori bounds.
Now, it remains to show that the set(58)Ω(Ϝ)={x∈C1-α(J,X):x=λϜ(x),λ∈(0,1)}
is bounded.
Letx∈Ω(Ϝ); then x=λϜ(x) for some λ∈(0,1). Similarly to the proof of Lemma 10, we obtain
(59)∥x∥C1-α≤κEα(Mρb).
This shows that the setΩ(Ϝ) is bounded.
As a result, by the conclusion of Theorem14, we obtain that Ϝ has a fixed point x. Therefore, system (1) has at least one mild solution on J. The proof is completed.


## 4. Optimal Control Results

In this section, we are concerned with the following Lagrange problem (P).Minimize a cost function of the form(60)𝒥(x,u):=∫0bℒ(t,x(t),u(t))dt,
among all the admissible state control pairs of the semilinear differential equations with Riemann-Liouville fractional derivatives (1); that is, find an admissible state control pair (x0,u0)∈C1-α(J,X)×Uad such that
(61)𝒥(x0,u0)≤𝒥(x,u),∀(x,u)∈C1-α(J,X)×Uad,
where x denotes the mild solution of system (1) corresponding to the control u∈Uad.For the existence of solution for problem (P), we will introduce the following assumptions:H(4):
the functionℒ:J×X×Y→R∪{∞} satisfies the following:(i)
the functionℒ:J×X×Y→R∪{∞} is Borel measurable;(ii)
ℒ(t,·,·) is sequentially lower semicontinuous on X×Y for almost all t∈J;(iii)
ℒ(t,x,·) is convex on Y for each x∈X and almost all t∈J;(iv)
there exist constantsc≥0, d>0, ψ is nonnegative, and ψ∈L1(J,R) such that
(62)ℒ(t,x,u)≥ψ(t)+c∥x∥X+d∥u∥Y.Next, we can give the following result on existence of optimal controls for problem (P).Theorem 16.
Let the assumptions of Theorem15 and H(4) hold. Then, Lagrange problem (P) admits an optimal pair; that is, there exists an admissible control pair (x0,u0)∈C1-α(J,X)×Uad such that
(63)𝒥(x0,u0)=∫0b‍ℒ(t,x0(t),u0(t))dt≤𝒥(x,u),ddddddddd∀(x,u)∈C1-α(J,X)×Uad.Proof.
Ifinf⁡{𝒥(x,u):(x,u)∈C1-α(J,X)×Uad}=+∞, then it is clear that the Lagrange problem (P) has an optimal pair.
Without loss of generality, we assume thatinf⁡{𝒥(x,u):(x,u)∈C1-α(J,X)×Uad}=ξ<+∞. Using H(4)(iv), we have ξ>-∞. By definition of infimum, there exists a minimizing sequence feasible pair {(xm,um)}⊂𝒫ad≡{(x,u):x that is a mild solution of system (1) corresponding to u∈Uad}, such that 𝒥(xm,um)→ξ as m→+∞. Since {um}⊆Uad(m=1,2,…), {um} is a bounded subset of the separable reflexive Banach space Lp(J,Y), there exists a subsequence, relabeled as {um}, and u0∈Lp(J,Y) such that
(64)um⇀u0inLp(J,Y).
SinceUad is closed and convex, by Mazur’s lemma, u0∈Uad.
Let{xm} denote the sequence of solutions of the system (1) corresponding to {um}; that is,
(65)xm(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)Bum(s)ds+∫0t‍(t-s)α-1Tα(t-s)f(s,xm(s))ds.
Now, we prove that {xm} is relatively compact on C1-α(J,X).
Firstly, it follows the boundedness of{um} and Lemma 10, and one can check that there exists a positive number ω such that ∥xm∥C1-α≤ω, which implies that ∥xm∥C1-α is uniformly bounded.
Next, denotegm(t)=t1-αxm(t); like the proof of Steps 3 and 4 in Theorem 15, {g:g=t1-αx(t)} is equicontinuous and relatively compact subsets of C(J,X). Hence, we can deduce that {xm} is relatively compact on C1-α(J,X). Therefore, there exists a function x0∈C1-α(J,X) such that
(66)xm⟶x0inC1-α(J,X).
Moreover, byH(1)(iii), we get
(67)∥f(t,xm(t))-f(t,x0(t))∥≤φ(t)∥xm-x0∥C1-α,
and then, in view of (66), we can obtain
(68)f(t,xm(t))⟶f(t,x0(t)),a.e.t∈J.
And by H(1)(ii), we get
(69)∥f(t,xm(t))∥≤ϕ(t)+ρt1-α∥x(t)∥≤ϕ(t)+ρω.
Thus, by applying the dominated convergence theorem, one can prove that(70)∫0t‍(t-s)α-1Tα(t-s)f(s,xm(s))ds⟶∫0t‍(t-s)α-1Tα(t-s)f(s,x0(s))ds,a.e.t∈J.
Similarly, we have(71)∫0t‍(t-s)α-1Tα(t-s)Bum(s)ds⟶∫0t‍(t-s)α-1Tα(t-s)Bu0(s)ds,a.e.t∈J.
Hence, it follows from (65) that
(72)x0(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)Bu0(s)ds+∫0t‍(t-s)α-1Tα(t-s)f(s,x0(s))ds;
that is, x0 denotes the sequence of solutions of the system (1) corresponding to u0.
Note thatH(4) implies that all of the assumptions of Balder (see Theorem 2.1 [27]) are satisfied. Hence, by Balder’s theorem, we can conclude that (x,u)→∫0b‍ℒ(t,x(t),u(t))dt is sequentially lower semicontinuous in the strong topology of L1(J,X)×L1(J,Y). Since Lp(J,X)×Lp(J,Y)⊂L1(J,X)×L1(J,Y), 𝒥 is also sequentially lower semicontinuous in Lp(J,X)×Lp(J,Y). Hence, 𝒥 is weakly lower semicontinuous on Lp(J,X)×Lp(J,Y), and since, by H(4)(iv), 𝒥>-∞, 𝒥 attains its infimum at (x0,u0)∈C1-α(J,X)×Uad; that is,
(73)ξ=lim⁡m→∞⁡∫0bℒ(t,xm(t),um(t))dt≥∫0b‍ℒ(t,x0(t),u0(t))dt=J(x0,u0)≥ξ.
The proof is completed.


## 5. An Example

Consider the following initial-boundary value problem of fractional parabolic control system with Riemann-Liouville fractional derivatives:(74)LDtαx(t,y)=∂2∂y2x(t,y)+e-t+t1-α(t+6)2sin⁡(x(t,y))+∫01‍q(y,τ)u(τ,t)dτ,ddddt∈J=[0,1],y∈[0,π],u∈Uad,x(t,0)=x(t,π)=0,t∈J=[0,1],I0+1-αx(t,y)|t=0=x0(y),t∈[0,1],y∈[0,π],
with the cost function
(75)𝒥(x,u)=∫01‍∫0π‍|x(t,y)|2dydt+∫01‍∫0π‍|u(t,y)|2dydt,
where α=2/3, q:[0,1]×[0,1]→R is continuous, and u∈L2(J,[0,π]).TakeX=Y=L2(J,[0,π]) and the operator A:D(A)⊂X→X is defined by
(76)Aω=ω′′,
where the domain D(A) is given by
(77){ω∈X:ω,ω′areabsolutelycontinuous,dω′′∈X,ω(0)=ω(π)=0}.
Then, A can be written as
(78)Aω=∑n=1∞‍n2(ω,ωn)ωn,ω∈D(A),
where ωn(x)=2/πsinnx(n=1,2,…) is an orthonormal basis of X. It is well known that A is the infinitesimal generator of a compact semigroup T(t)(t>0) in X given by
(79)T(t)x=∑n=1∞‍exp⁡-n2t(x,xn)xn,x∈X,∥T(t)∥≤e-1<1=M.We take the functionsu:Φx([0,π])→R as the controls, such that u∈L2(Φx([0,π])). It means that t→u(t) going from J into Y is measurable. Set U(t):={u∈Y:∥u∥Y≤ϑ}, where ϑ∈L2(J,R+). And restrict the admissible controls sets Uad to be all u∈L2(Φx([0,π])) such that ∥u(·,t)∥2≤ϑ(t), a.e. t∈J.Denote thatx(t,y)=x(t)(y); then
(80)f(t,x(t))(y)=e-t+t1-α(t+6)2sin(x(t))(y),B(t)u(t)(y)=[∫01‍q(τ)u(τ,t)dτ](y).It is easy to see that(81)∥f(t,x(t))∥=e-t+136t1-α∥x(t)∥:=ϕ(t)+ρt1-α∥x(t)∥,
and, for any x,y∈X,
(82)∥f(t,x(t))-f(t,y(t))∥≤1(t+6)2∥x-y∥C1-α:=φ(t)∥x-y∥C1-α.Hence, all the conditions of Theorem16 are satisfied, and the system (74) has an optimal pair solution.



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*Source: 216919-2014-04-15.xml*