Necessary and sufficient optimality conditions for optimization problems in function spaces and applications to control theory

We consider an abstract formulation for optimization problems in some Lp spaces. The variables are restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type. Second-order necessary and sufficient optimality conditions are established, where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces. The results are applied to an optimal control problem governed by a partial differential equation. Finally we compare the conditions obtained by applying this abstract procedure and those ones derived by using the methods adapted to the optimal control problem

Calculus of Tangent Sets and Derivatives of Set Valued Maps under Metric Subregularity Conditions

In this paper we intend to give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. A special attention is paid for the case of perturbation set-valued maps which appear naturally in optimization problems.Comment: 17 page

Preface

This volume contains the plenary lectures presented during the meeting of the MODE group of SMAI. The activity of the group is centered around optimization theory and practice, discrete mathematics, game theory, mathematical modeling in economics, finance and social sciences, nonlinear analysis, optimal control. The lectures presented during the industrial session and the short communications are not included in this volume. The editor would like to thank again the Scientific Committee, the Local Organizing Committee (and especially its Chairman Serge Gautier), Sponsors (Artelys, the Council of Pyrénées Atlantiques, the Council of the Aquitaine Region, the Council of the township of Pau, Electricité de France, the SMAI, Total-Fina- Elf, the University of Pau) and Exhibitors (Springer, World Scientific), which supported the organization of the conference and, of course, all participants