Asymptotic expansions for interior solutions of semilinear elliptic problems

J. Frederic Bonnans (1) , Francisco J. Silva (1, 2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems

In this work we consider the optimal control problem of a semilinear elliptic PDE with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated parametrized family of penalized problems, whose solutions define a central path converging to the solution of the original problem. Our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem. This approach allows us to obtain some specific error bounds in various norms and for a general class of barrier functions. In this manner, we generalize the results of the previous work which were obtained in the ODE framework.

Domains: Optimization and Control [math.OC]

J. Frederic Bonnans, Francisco J. Silva. Asymptotic expansions for interior solutions of semilinear elliptic problems. SIAM Journal on Control and Optimization, 2011, 49 (6), pp.2494-2517. ⟨inria-00436768⟩