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adjoint state p(u admits a unique analogous apply Theorem Ay(v Ay=f boundary conditions boundary value problems bounded set Chapter characterized closed constraints continuous linear form control is given control problems convex set convex subset cost function decoupling deduce defined denoted Dirichlet Problem dual elliptic operator equivalent everywhere in Q example exists a unique extract a subsequence formal calculations function is given Green's formula hence Hilbert space Hl(Q hold identity inequalities Inf J(v introduce the adjoint isomorphism kernel Lemma Let us assume Let us consider Lions Lions-Magenes method non-linear notation observation obtain optimal control parabolic operator partial differential equations preceeding Proof of Theorem properties Remark resp Riccati equations satisfies scalar product second order simultaneous solution solve system of equations systems governed Theorem 1.1 unilateral problem unique solution verify Vre^ad Vve%d weak topology weakly whence yo(x