Numerical approximation of nonconvex optimal control problems defined by parabolic equations

In this paper, we consider an optimal control problem for distributed systems governed by parabolic equations. The state equations are nonlinear in the control variable; the constraints and the cost functional are generally nonconvex. Relaxed controls are used to prove existence and derive necessary conditions for optimality. To compute optimal controls, a descent method is applied to the resulting relaxed problem. A numerical method is also given for approximating a special class of relaxed controls, notably those obtained by the descent method. Convergence proofs are given for both methods, and a numerical example is provided.