Remarks on the Controllability of Some Quasilinear Equations

TL;DR

This paper reviews controllability issues for nonlinear hyperbolic and parabolic equations and introduces a new unbounded perturbation technique for local controllability of quasilinear evolution equations.

Contribution

It presents a universal approach based on a novel unbounded perturbation technique for solving local controllability of quasilinear time-reversible equations.

Findings

New unbounded perturbation technique for controllability

Applicable to local existence and stabilization problems

Addresses key open problems in nonlinear hyperbolic and parabolic equations

Abstract

In this Note, we review the main existing results, methods, and some key open problems on the controllability of nonlinear hyperbolic and parabolic equations. Especially, we describe our recent universal approach to solve the local controllability problem of quasilinear time-reversible evolution equations, which is based on a new unbounded perturbation technique. It is also worthy to mention that the technique we develop can also be applied to other problems for quasilinear equations, say local existence, stabilization, etc.