A class of fractional evolution equations and optimal controls

Abstract

This paper concerns the existence of mild solutions for semilinear fractional evolution equations and optimal controls in the $\alpha$-norm. A suitable $\alpha$-mild solution of the semilinear fractional evolution equations is introduced. The existence and uniqueness of $\alpha$-mild solutions are proved by means of fractional calculus, singular version Gronwall inequality and Leray–Schauder fixed point theorem. The existence of optimal pairs of system governed by fractional evolution equations is also presented. Finally, an example is given for demonstration.

Introduction

The fractional differential equations have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, economics and science. We can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. [1], [2], [3], [4], [5], [6]. There has been a significant development in fractional differential equations. One can see the monographs of Kilbas et al. [7], Miller and Ross [8], Podlubny [9], Lakshmikantham et al. [10].

In this paper, we consider the following fractional evolution equations such as

$$\{\begin{array}{c}{D}^{q}x\left(t\right)=-Ax\left(t\right)+f(t,x\left(t\right))\text{,}t\in J=[0,T],q\in (0,1)\text{,}\hfill \\ x\left(0\right)=x_0\text{,}\hfill \end{array}$$

where $D^q$ is the Caputo fractional derivative of order $0<q<1$, $-A:D\left(A\right)\to X$ is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators $\{T\left(t\right),t\ge 0\}$, $f:J\times X_{\alpha }\to X$ is specified later, where $X_{\alpha }=D\left(A^{\alpha }\right)(0<\alpha <1)$ is a Banach space with the norm ${\Vert x\Vert }_{\alpha }=\Vert A^{\alpha }x\Vert$ for $x\in X_{\alpha }$.

In the last years, on the fractional differential equations in infinite dimensional spaces are attracted by many authors (see for instance [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24] and the references therein). When the fractional differential equations describe the performance index and system dynamics, an optimal control problem reduces to an optimal control problem. The fractional optimal control of a distributed system is an optimal control for which system dynamics are defined with fractional differential equations. There has been very little work in the area of fractional optimal control problem [25].

In order to obtain the existence of solutions for fractional differential equations, some authors use Krasnoselskii’s fixed point theorem or contraction mapping principle. It is obvious that the conditions for Krasnoselskii’s fixed point theorem are not easy to be verified sometimes and the conditions for contraction mapping principle are too strong.

In the present paper, we introduce a suitable $\alpha$-mild solution for system (1) which is different from the previous works such as [15], [19], [22], [23]. The new introduced $\alpha$-mild solution is associated with a probability density function and semigroup operator. Then we give some properties of two new linear operators associated with the probability density function and semigroup operator which are used throughout this paper. We prove the existence of $\alpha$-mild solutions for system (1) under some easy checked conditions. The main technique used here are fractional calculus, singular version Gronwall inequality visa Leray–Schauder fixed point theorem for compact maps. Further, we consider the Lagrange problem of systems governed by (1) and an existence result of optimal controls is presented. This results can be considered as a contribution to this emerging field.

The rest of this paper is organized as follows. In Section 2, we give some preliminary results on the fractional powers of the generator of an analytic compact semigroup and introduce the $\alpha$-mild solution of system (1). In Section 3, we study the existence and uniqueness of $\alpha$-mild solutions for system (1). In Section 4, we introduce a class of admissible controls and an existence result of optimal controls for a Lagrange problem (P) is proved. At last, an example is given to demonstrate the applicability of our result.