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a h(x applies approach APPROXIMATE MINIMIZATION C U C are uniformly computational constraint set control theory convergence questions converges to zero converges weakly convex Cullum Daniel difference equation constraints discretization fE example exist expanding fE f finite differences finite set functional f g r x h r x h(p x Hilbert space hypotheses lim inf h lim sup lower semicontinuous functional MINIMIZATION OF FUNCTIONALS minimization problem minimize h minimizing sequences n n n n nJ n n,o n,n n,i+l n,i NUMERICAL ANALYSIS NUMERICAL METHODS optimal control problems penalty function piecewise linear points of fp prob reflexive Banach space regularization or penalization satisfies n n n September 1970 sequentially compact subset sequentially lower semicontinuous set of points solve the MPC tends to zero Theorem 3.1 uniformly bounded unique solution vergence weak limit points weakly convergent weakly sequentially compact weakly sequentially lower x e C satisfies Yvon