Hamilton-Jacobi-Bellman approach for optimal control problems with discontinuous coefficients

Zhiping Rao 1, 2
1 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées
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 for Partial 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 associated Hamilton-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.
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Zhiping Rao. Hamilton-Jacobi-Bellman approach for optimal control problems with discontinuous coefficients. Optimization and Control [math.OC]. Ecole Polytechnique X, 2013. English. ⟨pastel-00927358⟩

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