The solvability problem for a controlled system with a fractional derivative and a causal operator

It is known that the contemporary approach in the theory of control systems and mathematical physics leads to models that are conveniently described by using differential equations and inclusions. Recently, the attention of many researchers (see [1]–[3] and the references therein) was attracted to generalizations of differential and functional differential equations and inclusions, namely to the class of functional equations and inclusions with causal operators. The term causal operator or Volterra operator in the sense of A. N. Tikhonov (see [4]), is used in mathematical physics to solve problems of differential equations, integro-differential equations, functional differential equations with finite or infinite delay, Volterra-type integral equations, functional equations of neutral type, etc. (see, for example, [5]). Papers [6]–[9] are devoted to the study of equations and inclusions with causal operators of various types, theorems on the existence of solutions, description of qualitative properties of solutions and various applications. At the same time, in recent decades, interest to the theory of fractional differential equations has increased significantly due to their effective applications in various fields of applied mathematics, physics, engineering, biology, economics, etc. (see, for example, monograph [10], articles [11]–[16], etc.). In [17]–[23] boundary value problems of various types for fractional differential equations and inclusions were considered. In this paper we develop the results of works [24]–[26] and consider a generalized boundary value problem for a feedback control system governed by a differential inclusion with a fractional derivative and a causal operator of the form
\(^CD^qx(t)∈Ax(t)+F(x)(t)+Bu(t)\), \(t∈[0,T]\), \(q∈(0;1)\),
satisfying the feedback condition
\(u∈Ψx\), \(u∈L^{\infty }([0,T];E)\)
and the general boundary condition:
\(Qx∈Sx\).
The paper has the following structure. The second section provides the necessary information from the theory of multivalued maps, measures of noncompactness, the concepts of a multivalued causal operator and a fractional derivative. In the next section, we study a system governed by a semilinear functional differential inclusion of fractional order, satisfying the feedback condition and the general boundary condition, and describe the properties of a multi-operator whose fixed points are solutions of this system. The practical significance of this work is contained in its applicability to the study of systems arising in various branches of natural sciences and governed by various classes of partial differential equations of a fractional order.

Keywords: controlled system, feedback, functional differential inclusion, fractional derivative, finite delay, measure of non-compactness, condensing operator, fixed point, topological degree.