Hamilton-Jacobi-Bellman approach for optimal control problems with discontinuous coefficients.
Abstract: This thesis deals with the Dynamical Programming and Hamilton-Jacobi-Bellman approach for a general class of deterministic optimal control problems with discontinuous coefficients.The tools essentially used in this work are based on the control theory, the viscosity theory forPartial Differential Equations, the nonsmooth analysis and the dynamical systems.
The first part of the thesis is concerned with the state constrained problem of discontinuous trajectories driven by impulsive dynamical systems. A characterization result of the value function of this problem has been obtained. Another contribution of this part consists of the extension of the HJB approach for the problems with time-measurable dynamical systems and in presence of time-dependent state constraints.
The second part is devoted to the problem on stratified domain, which consists of a union of subdomains separated by several interfaces. One of the motivations of this work comes from the hybrid control problems. Here new transmission conditions on the interfaces have been obtained to ensure the uniqueness and the characterization of the value function.
The third part investigates the homogenization of Hamilton-Jacobi equations in the framework of state-discontinuous Hamiltonians. This work considers the singular perturbation of optimal control problem on a periodic stratified structure. The limit problem has been analyzed and the associatedHamilton-Jacobi equation has been established. This equation describes the limit behavior of the value function of the perturbed problem when the scale of periodicity tends to 0.
Keywords: Optimal control problems, Hamilton-Jacobi-Bellman equations, viscosity solutions,nonsmooth analysis, impulsive differential equations, state constraints, multi-domains, stratified dynamical system, transmission conditions, singular perturbation.
Understanding large text corpora via sparse optimization.
Contrôle optimal d’équations différentielles avec – ou sans – mémoire.
Analyse de sensibilité pour des problèmes de commande optimale ; commande optimale stochastique sous contrainte en probabilité.
An approximation scheme for finding equilibria in energy markets with risk aversion.
Abstract: We discuss some recent results on convergence and rate of convergence of the classical augmented Lagrangian algorithm. First, we prove local primal-dual convergence under the sole assumption that the dual starting point is close to a multiplier satisfying the second-order sufficient optimality condition. In particular, no constraint qualifications of any kind are needed. Previous literature on the subject required, in addition, the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second-order sufficient condition. Second, we show that when applied to optimization problems with complementarity constraints (MPCC), the augmented Lagrangian approach has theoretical global convergence guarantees that compare favorably to the alternatives, whether standard nonlinear programming solvers or those specially designed for this problem class. Moreover, we show that in practice the ALGENCAN implementation of the augmented Lagrangian method is a very good choice for MPCC if robustness and the quality of computed solutiuon is of primary concern.
Modelisation de la croissance metastatique d’un cancer et des traitements anti-cancereux.
Abstract: Le cancer est devenu la première cause de mortalité en France. L’utilisation de modèles mathématiques pour décrire cette maladie ainsi que les traitements administrés semble très prometeuse. Nous verrons dans cet exposé un certain nombre de difficultés auxquelles doivent faire face le médecin et les outils que les mathématiciens peuvent leur apporter.
(Sup + Bolza)-control problems as dynamic differential games.
Abstract: We consider a (L∞+Bolza)-control problem, namely a problem where the payoff is the sum of a sup (actually ess-sup) functional and a classical Bolza functional. Owing to the ⟨L1,L∞⟩ duality, the (L∞+Bolza)-control problem is rephrased in terms of a static differential game, where a new variable k plays the role of a maximizer. In this framework 1 − k is regarded as the available fuel for the maximizer. The relevant (and unusual) fact is that this static game is equivalent to the corresponding dynamic differential game, which allows the (upper) value function to verify a rather simple boundary value problem.
The whole approach will be hopefully utilized to get necessary conditions for a curve to be a minimizer.
Calcul d’ensemble de trajectoires de systèmes hybrides par des méthodes numériques garanties.
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An overview of scalable control and estimation using max-plus methods.
Abstract: Conventional approaches to solving optimal control problems using dynamic programming require the solution of Hamilton Jacobi equations. Numerical solution techniques suffer from the curse of dimensionality and are thus intractable in higher dimensions. This talk will provide a gentle introduction to a promising class of control synthesis methods which are more scalable than grid based methods. Further the application of these ‘max-plus’ methods to an intriguing range of problems in estimation and control of linear, non-linear and multi-sensor problems will be described.
Explicit solution for singular calculus of variations in infinite horizon.
Abstract: We consider a one dimensional infinite horizon calculus of variations problem (P), where the integrand is linear with respect to the velocity. The Euler-Lagrange equation, when defined, is not a differential equation as usual, but reduces to an algebraic (or transcendental) equation $C(x)=0 $. Thus this first order optimality condition is not informative for optimal solutions with initial condition $x_0 $ such that $C(x_0) \neq 0 $. To problem (P) we associate an auxiliary calculus of variations problem whose solutions connect as quickly as possible the initial conditions to some constant solutions. Then we deduce the optimality of these curves, called MRAPs (Most Rapid Approach Path), for (P). According to the optimality criterium we consider, we have to assume a classical transversality condition. We observe that (P) possesses the Turnpike property, the Turnpike set being given by the preceding particular constant solutions of the auxiliary problem.
Pursuit-Evasion Games with Multi-Pursuer: a decomposition approach.
Abstract: In a pursuit-evasion game, each pursuer attempts to minimize the distance between himself (P) and the evader (E) and capture it in the shortest time, whereas the evader tries to maximize the same distance to escape from being captured. In this talk, we deal with PE games with an evader and n pursuers. It is easy to imagine that the dimension of the space where the agents are moving increases with the number of pursuers, making the problem hard to solve. We will use the well-known PDE approach and an original decomposition technique to get over the big dimensionality contained in the problem. Simulation results will show the effectiveness of the proposed strategy of resolution as well as the limitations of it.
Optimal control of single species biological population.
Stabilité des arcs singuliers pour un problème de commande optimale perturbé.
Convergence of one-step methods for SDEs.
Modélisation de la production des globules rouges : Équations de transport non-linéaires et stabilité.