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: ∂y(/t)/∂t + Ay(t)=f(t,y(t)) + u(t), t ε [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 arising from the controllability constraint and y_n^∗ is the solution of the parabolic non-linear system defined 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.