Asymptotic expansion for the solution of a penalized control constrained semilinear elliptic problems
Francisco J. Silva, J. Frédéric Bonnans · 2009
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 family of penalized problems, parametrized by e > 0, 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 [2] which were obtained in the ODE framework. Key-words: Optimal control of PDE, interior-point algorithms, control constraints, expansion of solutions. ∗ INRIA-Saclay and CMAP, Ecole Polytechnique, 91128 Palaiseau, France ([email protected]) † INRIA-Saclay and CMAP, Ecole Polytechnique, 91128 Palaiseau, France ([email protected]) Developpement asymptotique de la solution d’un probleme de commande optimale semi lineaire elliptique penalise Resume : Dans ce travail nous considerons le probleme de commande optimale d’une equation semi lineaire elliptique avec conditions de Dirichlet homogene au bord, la commande etant distribuee sur le domaine et positive. L’approche est de considerer une famille de problemes penalises par e > 0, dont la solution definit une trajectoire centrale qui converge vers la solution du probleme original. Notre but est d’obtenir un developpement asymptotique de la solution du probleme penalise au voisinege de la solution du probleme original. Notre approche nous permet d’obtenir des estimations d’erreur dans differentes normes et pour une classe generale de fonctions barriere. Ceci etend les resultats de [2], obtenus dans un cadre de commande optimale d’equations differentielles. Mots-cles : Commande optimale des EDP, algorithmes de points interieurs, contraintes sur la commande, d’eveloppement des solutions. Asymptotic expansions for interior solutions of semilinear elliptic problems 3