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adjoint Algorithm application approximation assume assumptions Banach space BFGS Method boundary conditions boundary value problem bounded Compute cone consider continuous continuous embeddings convergence convex convex cone corresponding curl defined denote derivative dimensional Dirichlet discrete distributed parameter systems domain dual dual space element elliptic estimates exact controllability problem exists a constant finite energy finite-dimensional Fourier function given Hence Hessian Hilbert space hypotheses inequality iteration Lasiecka Lemma linear Lipschitz Lipschitz continuous Math matrix minimization Moreover nodes nonlinear norm obtain operator optimal control problems optimal problem optimality conditions parabolic partial differential equations perturbations positive definite Proof properties prove quasi-Newton methods reachable set reduced SQP method regularity Remark resp respect result satisfies second order semigroup semilinear sequence Shape gradient Shape Hessian solve stability subset surjective Theorem 2.1 theory topology Triggiani unique solution variation vector velocity fields verified wave equation weak Zolesio