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Hamilton Jacobi Bellman Approach for some applied optimal control problems.

Abstract : The main objective of this thesis is to analyze the Hamilton Jacobi Bellman approach for some control problems of unusual forms. The first work is devoted to the numerical approximations of unbounded and discontinuous value functions associated with some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). The proof is based on classical shaking and regularization techniques. The second contribution concerns the probabilistic reachablility analysis. In particular, we characterize the chance-constrained backward reachable set by a level set of a discontinuous value function and we use the first theoretical results to derive the corresponding error estimates. In the second part of this thesis, we study a class of state constrained optimal control problem with maximum cost. We first describe the epigraph of the value function by an auxiliary optimal control problem whose value function is Lipschitz continuous. We show that the new value function is the unique Lipschitz continuous viscosity solution of a Hamilton Jacobi equation with a Dirichlet condition. Here, we give a review of the optimal trajectories and the associated feedback control for such control problems. In particular, we prove the convergence of a sequence of approximated optimal trajectories to the continuous one. We establish a link between the control problem and a viability kernel associated with an exit time function. The obtained results for the state constrained control problem with maximum cost are then extended to the state constrained control problem with Bolza cost. The study is motivated by a real application: the abort landing during low altitude wind-shears. Many algorithms of reconstruction of optimal feedback trajectories are studied and compared from numerical and theoretical points of view.
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Contributor : Mohamed Assellaou <>
Submitted on : Wednesday, February 17, 2016 - 10:20:15 PM
Last modification on : Thursday, March 5, 2020 - 6:15:28 PM
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  • HAL Id : tel-01275676, version 1


Mohamed Assellaou. Hamilton Jacobi Bellman Approach for some applied optimal control problems.. Mathematics [math]. Ensta ParisTech, 2015. English. ⟨NNT : 2015ESTA0022⟩. ⟨tel-01275676⟩



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