Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

TL;DR

This paper establishes conditions for exact local controllability of parabolic evolution equations to eigensolutions using bilinear control, with explicit constants and applications to low-dimensional parabolic PDEs.

Contribution

It provides a constructive method with explicit constants for controllability to eigensolutions in parabolic equations via bilinear control, extending previous results.

Findings

Sufficient conditions for local controllability to eigensolutions.

Explicitly computable constants in controllability results.

Applications to low-dimensional parabolic equations.

Abstract

In a separable Hilbert space $X$, we study the controlled evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A\geq-\sigma I$ ($\sigma\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the $j$th eigensolution for any $j\geq1$. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.