Recently, a generalized Caputo-Katugampola fractional derivative was proposed in [2] by Katugampola and afterwards he provided the existence of solutions for fractional differential equations defined by this derivative. This Caputo-Katugampola fractional derivative extends the well known Caputo and Caputo-Hadamard fractional derivatives into a single form. Even if Katugampola fractional integral operator is an Erdelyi-Kober type operator it is argued in [2] that is not possible to derive Hadamard equivalence operators from Erdelyi-Kober type operators. Also, in some recent papers, several qualitative properties of solutions of fractional differential equations defined by Caputo-Katugampola derivative were obtained.

In this paper we study the following problem

D_C^α,ρ x(t) ÎF(t,x(t)) a.e. ([0,T]),  x(0)ÎX_0, x'(0)Î X_1,                    (1)

where αÎ (1,2], ρ≥1, D_C^α,ρ is the Caputo-Katugampola fractional derivative, F:[0,T]×R®Ƥ(R) is a set-valued map and X_0,X_1 are closed sets in R.

Consider S the set of all solutions of (1) and let R(T) be the reachable set of (1) at moment T. For a solution z(.)ÎS and for a locally Lipschitz function h:R®R^m we say that the differential inclusion (1) is h-locally controllable around z(.) if h(z(T))Îint(h(R(T))). In particular, if h is the identity mapping and m=1 the above definitions reduces to the usual concept of local controllability of systems around a solution.

Our aim is to obtain a sufficient condition for h-local controllability of inclusion (1). This result is derived using a technique developed by Tuan for classical differential inclusions ([3]). More exactly, we show that inclusion (1) is h-locally controlable around the solution z(.) if a certain linearized fractional differential inclusion is l-locally controlable around the null solution for every lζ h(z(T)), where ¶h(.) denotes Clarke's generalized Jacobian of the locally Lipschitz function h(.). The main tools in our approach is a continuous version of Filippov's theorem for solutions of problem (1) obtained recently in [1] and a certain generalization of the classical open mapping principle due to Warga.

References

1. A. Cernea, Continuous family of solutions for fractional integro-differential inclusions of Caputo-Katugampola type, Progress Fractional Diff. Appl., 5, 1-6 (2019).

2. U. N. Katugampola, A new approach to generalized fractional derivative, Bull. Math. Anal. Appl., 6, 1-15 (2014).

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