DEGENERATE DISTRIBUTED CONTROL SYSTEMS WITH FRACTIONAL TIME DERIVATIVE

Marina V Plekhanova

doi:10.15826/umj.2016.2.006

Abstract

The existence of a unique strong solution for the Cauchy problem to semilinear nondegenerate fractional differential equation and for the generalized Showalter–Sidorov problem to semilinear fractional differential equation with degenerate operator at the Caputo derivative in Banach spaces is proved. These results are used for search of solution existence conditions for a class of optimal control problems to a system described by the degenerate semilinear fractional evolution equation. Abstract results are applied to the research of an optimal control problem solvability for the equations system of Kelvin–Voigt fractional viscoelastic fluids.

Highlights

Let X, Y be Banach spaces, L, M : X → Y be linear operators, ker L= {0}, α > 0, m ∈ N, m − 1 < α ≤ m, r ∈ {0, 1, . . . , m − 1}, N : (t0, T ) × X r+1 → Y
The equation with left-hand side in the form DαLx is considered. It has different properties beginning with the definition of a solution
The solvability in the sense of the classical solution for another class of degenerate fractional equations (0.1) in Banach spaces with restriction on the image of N was studied in [10]

Summary

Introduction

Let X , Y be Banach spaces, L, M : X → Y be linear operators, ker L= {0}, α > 0, m ∈ N, m − 1 < α ≤ m, r ∈ {0, 1, . . . , m − 1}, N : (t0, T ) × X r+1 → Y. The main purpose of the paper is to study the initial value problems unique solvability to the fractional order differential equation. In the sense of the strong solutions and the solvability of optimal control problems for systems with the state that described by (0.1) Such equations are called degenerate because of degeneracy of the operator L at the highest derivative. The solvability in the sense of the classical solution for another class of degenerate fractional equations (0.1) in Banach spaces with restriction on the image of N was studied in [10]. In contrast to the mentioned works the results of the present paper concern the existence of a unique strong solution for semilinear degenerate evolution fractional order equations that previously were not investigated. A research of control problems for semilinear degenerate evolution equations that has previously not been studied is presented

Nondegenerate linear equation of fractional order

The Cauchy problem for the semilinear equation

Degenerate semilinear equation

Optimal control problem

Optimal control for fractional Kelvin—Voigt fluid