On the Optimal Control Problem of Laser Surface Hardening

We discuss an optimal control problem of laser surface hardening of steel which is governed by a dynamical system consisting of a semilinear parabolic equation and an ordinary differential equation with a non differentiable right hand side function f+. To avoid the numerical and analytic difficulties posed by f+, it is regularized using a monotone Heaviside function and the regularized problem has been studied in literature. In this article, we establish the convergence of solution of the regularized problem to that of the original problem. The estimates, in terms of the regularized parameter, justify the existence of solution of the original problem. Finally, a numerical experiment is presented to illustrate the effect of regularization parameter on the state and control errors.