The cost of controlling degenerate parabolic equations by boundary controls

TL;DR

This paper analyzes the controllability of a one-dimensional degenerate parabolic equation with boundary control at the degeneracy point, extending the moment method and deriving bounds on control costs as degeneracy increases.

Contribution

It extends the moment method to degenerate parabolic equations and provides bounds on controllability costs depending on the degeneracy parameter.

Findings

Reachable targets characterized using extended moment method.

Control cost blows up as degeneracy parameter approaches 1.

Optimal bounds on control cost with respect to degeneracy parameter.

Abstract

We consider the one-dimensional degenerate parabolic equation $$ u_t - (x^\alpha u_x)_x =0 \qquad x\in(0,1),\ t \in (0,T) ,$$ controlled by a boundary force acting at the degeneracy point $x=0$. First we study the reachable targets at some given time $T$ using $H^1$ controls, extending the moment method developed by Fattorini and Russell to this class of degenerate equations. Then we investigate the controllability cost to drive an initial condition to rest, deriving optimal bounds with respect to $\alpha$ and deducing that the cost blows up as $\alpha \to 1^-$.