Abstract

This paper is concerned with the approximate controllability of Sobolev-type (k,ψ)−Hilfer fractional differential equations (FDEs) with control and Sobolev-type (k,ψ)−Hilfer fractional initial conditions in Hilbert spaces. By means of two operators kSψα,β, kTψα and the k−probability density function, the definition of mild solutions for the studied problem was given. Then, via (k,ψ)−Hilfer fractional derivative and by combining the techniques of fractional calculus and the fixed point theorem, we analyzed the existence and uniqueness of mild solutions. With the help of a Cauchy sequence and approximate techniques, we established some sufficient conditions for the approximate controllability of the proposed control system. Finally, an example is presented for the demonstration of obtained results.

Issue Section:
Research Papers
Keywords:
approximate controllability, Sobolev-type equation, fractional differential equations, (k,ψ)−Hilfer derivative, Banach's fixed point theorem, mild solutions
Topics:
Control systems, Density, Differential equations, Probability, Theorems (Mathematics), Space

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