Second order optimality conditions and their role in PDE control

If f : Rn R is twice continuously differentiable, f’(u) = 0 and f’’(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order suffcient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled order sufficient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled? It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where f’’(u) exists can be useless to ensure positive definiteness of the quadratic form v f’’(u)v2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the theory is developed for problems in function spaces with simple box constraints of the form a = u = ß. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of f’’(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense. As a first application of second-order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711, the second author by DFG in the framework of the Collaborative Research Center SFB 910, project B6

Line shape and Doppler broadening in resonant CARS and related nonlinear processes through a diagrammatic approach

We examine the effect of electronic resonance enhancement on the line broadening for Coherent Anti-Stokes Raman Scattering (CARS) and related third-order non linear processes, in low pressure gases. These phenomena are analysed theoretically within the framework of a time-ordered diagrammatic representation of nonlinear polarizations. Diagrams facilitate the classification of the various physical processes and provide a way to readily derive all the susceptibility contributions. In CARS, the susceptibility terms which are Doppler-free and thus prevail in the Doppler limit are different from those which dominate in the collision broadening regime. This causes important changes in the spectral content. On the contrary, the same terms are seen to dominate, in both regimes, in coherent Stokes Raman scattering or in stimulated Raman scattering. Typical line contours are displayed. The generalization to all nonlinear processes is outlined.Nous étudions l'influence de la résonance électronique sur les largeurs de raies de la Diffusion Raman Anti-Stokes Cohérente (DRASC) dans les gaz à faible pression. D'autres processus non linéaires du 3e ordre sont aussi examinés. Nous avons effectué une étude théorique de tous ces phénomènes en utilisant une représentation diagrammatique des polarisations non linéaires. Les diagrammes permettent de classer aisément les différents processus physiques qui contribuent à la création de la polarisation non linéaire, et de calculer rapidement grace à des règles simples les termes correspondants de la susceptibilité. En DRASC, les termes de la susceptibilité qui dominent en régime Doppler (parce qu'ils ne subissent pas d'élargissement Doppler) diffèrent de ceux qui dominent en régime collisionnel. Ceci entraîne d'importantes différences entre les contenus spectraux des régimes Doppler et collisionnel. Au contraire, en diffusion Stokes cohérente ainsi qu'en diffusion Raman stimulée, ce sont les mêmes termes qui dominent dans les deux régimes. Nous traçons quelques formes de raies types. Nous montrons qu'il est aisé de généraliser cette étude à tous les processus non linéaires