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The current paper initially studies the optimal control of linear$\psi$-Hilfer fractional derivatives with state-dependent control constraintsand optimal control for a particular type of cost functional. Then, weinvestigate the approximate controllability of the abstract fractionalsemilinear differential inclusion involving $\psi$-Hilfer fractional derivativein reflexive Banach spaces. It is known that the existence, uniqueness, optimalcontrol, and approximate controllability of fractional differential equationsor inclusions have been demonstrated for a similar type of fractionaldifferential equations or inclusions with different fractional order derivativeoperators. Hence it has to research fractional differential equations with moregeneral fractional operators which incorporate all the specific fractionalderivative operators. This motivates us to consider the $\psi$-Hilferfractional differential inclusion. We assume the compactness of thecorresponding semigroup and the approximate controllability of the associatedlinear control system and define the control with the help of duality mapping.We observe that convexity is essential in determining the controllabilityproperty of semilinear differential inclusion. In the case of Hilbert spaces,there is no issue of convexity as the duality map becomes simply the identitymap. In contrast to Hilbert spaces, if we consider reflexive Banach spaces,there is an issue of convexity due to the nonlinear nature of duality mapping.The novelty of this paper is that we overcome this convexity issue andestablish our main result. Finally, we test our outcomes through an example.

Optimal Control and Approximate controllability of fractional semilinear differential inclusion involving $\psi$- Hilfer fractional derivatives