Periodic Hamiltonian systems in shape optimization problems with Neumann boundary conditions

Abstract

The recent implicit parametrization theorem, based on simple Hamiltonian systems, allows the description of domains and their boundaries and, consequently, it provides a general fixed domain approximation method in shape optimization problems, using optimal control theory. Here, we discuss topology and shape optimization in the difficult case of Neumann boundary conditions, with a combined cost including both distributed and boundary observation. We give an unexpected general equivalence property with constrained optimal control problems, preserving differentiability. An important new ingredient in the arguments is the differentiability of the period for the Hamiltonian systems, with respect to functional variations. Due to the differentiability properties, we can use descent algorithms of gradient type. In the experiments, our approach can modify the topology both by closing holes or by creating new holes. We underline the applicability of this new methodology to large classes of shape optimization problems.