On approximation theorems for controllability of non-linear parabolic problems

Abstract

In this paper, we consider the following control system governed by the non-linear parabolic differential equation of the form: partial derivative(t)/partial derivative t +Ay(t)=f(t,y(t))+u(t), t epsilon[0, T], y(0) =y0, where A is a linear operator with dense domain and f (t, y) is a non-linear function. We have proved that under Lipschitz continuity assumption on the non-linear function f (t, y), the set of admissible controls is non-empty. The optimal pair (u*, y*) is then obtained as the limit of the optimal pair sequence {(u(n)*, y(n)*)}, where u(n)* is a minimizer of the unconstrained problem involving a penalty function aris. n n ing from the controllability constraint and y(n)* is the solution of the parabolic non-linear system defined n above. Subsequently, we give approximation theorems which guarantee the convergence of the numerical schemes to optimal pair sequence. We also present numerical experiment which shows the applicability of our result.