The existence of mild solutions and approximate controllability for Riemann–Liouville fractional neutral evolution systems with nonlocal conditions of a fractional order is investigated. The Laplace transform and semigroup theory are the tools used to prove the existence. In turn, approximate controllability is proved on the basis of a Nemytskii operator, a Mittag-Lefﬂer function and certain hypotheses using ﬁxed point theorems, as well as the construction of a Cauchy sequence. An example is provided to highlight the main results.

The existence of mild solutions and approximate controllability for nonlinear fractional neutral evolution systems

Abstract In this work, we are concerned with the approximate controllability of fractional nonlinear differential equations with nonlocal conditions of order in Banach spaces. As far as we know, few articles are investigating this issue. The idea is to see under which sufficient conditions the proposed control problem is approximately controllable. The discussion is based on the theory of resolvent operator, fractional calculus techniques, and Krasnoselskii's fixed point theorem under the assumption that the associated linear system is approximately controllable. The obtained results improve some existing analogous ones on this topic to some extent. Finally, an example is provided to illustrate the applications of the obtained results. 

pdf/approximate-controllability-of-fractional-differential-2gx67onpafe7.pdf

Approximate controllability of fractional differential systems with nonlocal conditions of order q∈(1,2)$$ q\in \left(1,2\right) $$ in Banach spaces

We investigate a class of fractional neutral evolution equations on Banach spaces involving Caputo derivatives. Main results establish conditions for the controllability of the fractional-order system and conditions for existence of a solution to an optimal control problem of minimum energy. The results are proved with the help of fixed-point and semigroup theories.

/pdf/minimum-energy-problem-in-the-sense-of-caputo-for-fractional-1b17adsg.pdf

Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces

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Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces