Approximate controllability of nonlocal fractional integrodifferential control systems of order $1 < \alpha < 2$

Abstract

In this article, we obtain sucient conditions for the approximate controllability of the fractional integral-differential systemCD0^{\alpha} x (t) = Ax (t) + Bu (t) + I0^{2-\alpha} f (t; x (t) ;Hx (t)) ; t \in  2 (0; b];x (0) + g0 (x) = x0 \in X,    x'(0) + g1 (x) = x1 \in X;where A : D(A) \subseteq X  \to X is sectorial operator on a Hilbert space X, Bis a bounded linear operator from admissible Hilbert control space U intoX. The nonlinear function f : JxXxX \to X, and the nonlocal functionsg0 and g1 are continuous functions. The operator H is bounded on X.