Finite-approximate controllability of evolution systems via resolvent-like operators

TL;DR

This paper extends variational methods to establish finite-approximate controllability of semilinear evolution systems in Hilbert spaces, providing new characterizations and control strategies with applications to heat equations.

Contribution

It introduces a resolvent-like operator characterization for controllability and develops conditions for finite-approximate controllability of semilinear systems.

Findings

Characterization of controllability via resolvent-like operators

Conditions ensuring finite-approximate controllability

Application to heat equations

Abstract

In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability ( finite-approximate controllability) for the semilinear evolution equations in Hilbert spaces. We state a useful characterization of the finite-approximate controllability for linear evolution equation in terms of resolvent-like operators. We also find a control so that, in addition to the approximate controllability requirement, it ensures finite dimensional exact controllability. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the finite-approximate controllability of the semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated.