A Caputo type fractional derivative of a function with respect to another function that extends and unifies several fractional derivatives existing in the literature like Caputo, Caputo-Hadamard, Caputo-Katugampola (e.g., [1]) was intensively studied in recent years. Existence results and qualitative properties of the solutions for fractional differential equations defined by this fractional derivative are obtained.

In this paper we study a Cauchy problem associated to a fractional differential inclusion defined by this derivative. Our aim is to obtain a sufficient condition for h-local controllability of this problem. This result is derived using a technique developed by Tuan for classical differential inclusions ([3]). More exactly, we show that the original fractional differential inclusion is h-locally controlable around a reference solution z(.) if a certain linearized fractional differential inclusion is g-locally controlable around the null solution for every g(.) that belongs to Clarke's generalized Jacobian of the locally Lipschitz function h(.). The main tools in our approach are a continuous version of Filippov's theorem for solutions of this problem obtained recently in [2] and a certain generalization of the classical open mapping principle due to Warga ([4]).

References

1.R. Almeida, A.B. Malinowska and M.T.T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Meth. Appl. Sci., 41 (2018), 336-352.

2. A. Cernea, Continuously parametrized solutions of a fractional integro-differential inclusion, Applied Anal. Optim., 5 (2021), 157-167

3.  H. D. Tuan, On controllability and extremality in nonconvex differential inclusions,  J. Optim. Theory Appl., 85 (1995), 437-474.

4. J. Warga, Controllability, extremality and abnormality in nonsmooth optimal control, J. Optim. Theory Appl., 41 (1983), 239-260.