OPTIMALITY CONDITIONS AND LAGRANGE MULTIPLIERS FOR SHAPE AND TOPOLOGY OPTIMIZATION PROBLEMS

We discuss first order optimality conditions for geometric optimization prob-lems with Neumann boundary conditions and boundary observation. The methods we develop here are applicable to large classes of state systems or cost functionals. Our ap-proach is based on the implicit parametrization theorem and the use of Hamiltonian systems. It establishes equivalence with a constrained optimal control problem and uses Lagrange multipliers under a simple constraint qualification. In this setting, general functional varia-tions are performed, that combine classical topological and boundary variations, in a natural way.