We consider an optimal control problem involving semilinear parabolic partial differential equations, with control constraints. Since no convexity assumptions are made, the problem is reformulated in relaxed form The state equation is discretized using a finite element method in space and a Crank-Nicolson method in time, while the controls are approximated by blockwise constant relaxed controls The first result is that accumulation points of discrete optimal (resp extremal) controls are optimal (resp extremal) for the continuous relaxed problem We also propose a conditional gradient method for solving each discrete problem, and a progressively refining discrete conditional gradient method, both generating discrete relaxed controls, for solving the continuous relaxed problem The second method has the advantage of reducing computations and memory Relaxed controls computed by these methods can then be simulated by piecewise constant classical controls using a simple approximation procedure Finally, a numerical example is given.
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