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

#### **Dumitru Baleanu 1,2, Vladimir E. Fedorov 3,4,\*, Dmitriy M. Gordievskikh 5 and Kenan Ta¸s 1**


Received: 1 June 2019; Accepted: 9 August 2019; Published: 12 August 2019

**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.

**Keywords:** approximate controllability; degenerate evolution equation; fractional Caputo derivative; sectorial operator

**MSC:** 93B05; 35R11; 34G99
