Approximate Controllability of the System Governed by Double Coupled Semilinear Degenerate Parabolic Equations

Abstract

This paper concerns the approximate controllability of the initial-boundary problem of double coupled semilinear degenerate parabolic equations. The equations are degenerate at the boundary, and the control function acts in the interior of the spacial domain and acts only on one equation. We overcome the difficulty of the degeneracy of the equations to show that the problem is approximately controllable in $L^2$ by means of a fixed point theorem and some compact estimates. That is to say, for any initial and desired data in $L^2$, one can find a control function in $L^2$ such that the weak solution to the problem approximately reaches the desired data in $L^2$ at the terminal time.