Error estimates for the Euler discretization of an optimal control problem with first-order state constraints

Joseph Frederic Bonnans 1, 2 Adriano Festa 3
1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CNRS - Centre National de la Recherche Scientifique : UMR7641, X - École polytechnique, UMA - Unité de Mathématiques Appliquées, Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 RICAM - Johann Radon Institute for Computational and Applied Mathematics
Abstract : We study the error introduced in the solution of an optimal control problem with first order state constraints, for which the trajectories are approximated with a classical Euler scheme. We obtain order one approximation results in the L ∞ norm (as opposed to the order 2/3 obtained in the literature). We assume either a strong second order optimality condition, or a weaker one in the case where the state constraint is scalar, satisfies some hypotheses for junction points, and the time step is constant. Our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems.
Keywords : Nonlinear systems Optimal control State constraints Rate of convergence Euler discretization
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SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2017, 55 (2), pp.445--471
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Joseph Frederic Bonnans, Adriano Festa. Error estimates for the Euler discretization of an optimal control problem with first-order state constraints. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2017, 55 (2), pp.445--471. 〈hal-01093229v2〉

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