Finite-Approximate Controllability of Riemann–Liouville Fractional Evolution Systems via Resolvent-Like Operators

Abstract

This paper presents a variational method for studying approximate controllability and infinite-dimensional exact controllability (finite-approximate controllability) for Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. A useful criterion for finite-approximate controllability of Riemann–Liouville fractional linear evolution equations is formulated in terms of resolvent-like operators. We also find that such a control provides finite-dimensional exact controllability in addition to the approximate controllability requirement. Assuming the finite-approximate controllability of the corresponding linearized RL fractional evolution equation, we obtain sufficient conditions for finite-approximate controllability of the semilinear RL fractional evolution equation under natural conditions. The results are a generalization and continuation of recent results on this subject. Applications to fractional heat equations are considered.