Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

Abstract

We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter-Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input. Mathematics 2019, 7, 735 2 of 15 homogeneous ( f ≡ 0) Equation (1). The existence of a unique solution of the Cauchy problem and of the Showalter-Sidorov problem

Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems

“…Here we used the theory of the degenerate evolution equations (see works [2,20,21]). In the fourth section the result on the existence of a unique strong solution for problem (1), (2) is applied to the proof of the optimal control existence for (1)- (4). The last section of the paper illustrates the obtained abstract results on an example of an initial-boundary value problem for a partial differential equation.…”

“…We are going to establish solvability conditions of problem (1)- (4). In recent decades, fractional integro-differential calculus has become one of the most important tools for solving mathematical modeling problems [4][5][6][7][8]. On the other hand various issues of the control theory, including unique solvability, are of interest to many authors.…”

A study on controllability of impulsive fractional evolution equations via resolvent operators

“…In recent years, the controllability problems for various linear and nonlinear deterministic and stochastic dynamic systems have been studied in many publications using different method, we refer the reader to [2,6,24]. In addition, Kailasavalli et al [14] acknowledged the existence and controllability of fractional neutral integro-differential systems with SDD with Banach contraction and resolvent operator technique as the main reference.…”

“…On the other hand, in recent years, much attention has been paid to establishing sufficient conditions for the controllability of linear fractional dynamical systems of order 0 < α < 1 by several authors; see a recent monograph [1,2,4,5,8,22,23] and various papers [2,6]. However, there is no work that reported on the problem of controllability of nonlinear fractional dynamical system of order 1 < α < 2, to the best of our knowledge, up until now the controllability for a class of impulsive fractional integro-differential evolution equation with fractional derivative of order α ∈ (1, 2] has not been investigated in the literature.…”

A Class of Fractional Degenerate Evolution Equations with Delay

“…Integer or fractional-order degenerate differential equations, i.e., evolution equations not solved with respect to the highest order derivative, are often used to describe various processes in science and engineering: in [9,10] certain classes of the time-fractional order partial differential equations with polynomials differential with respect to the spatial variables elliptic self-adjoint operator, which contain some equations from hydrodynamics and the filtration theory, are studied. In [11] approximate controllability issues for such models are investigated; the unique solvability of similar equations with distributed order time derivatives are researched in [12].…”