Nonlinear optimal control: a numerical scheme based on occupation measures and interval analysis

This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is writt...

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Published in:Computational optimization and applications Vol. 77; no. 1; pp. 307 - 334
Main Authors: Delanoue, Nicolas, Lhommeau, Mehdi, Lagrange, Sébastien
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2020
Springer Nature B.V
Springer Verlag
Subjects:
Convex and Discrete Geometry
Fineness
Interval arithmetic
Linear programming
Lower bounds
Management Science
Mathematical programming
Mathematics
Mathematics and Statistics
Nonlinear control
Operations Research
Operations Research/Decision Theory
Optimal control
Optimization
Optimization and Control
Statistics
Nonlinear optimal control
Interval arithmetic
Continuous programming
Optimization
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-020-00198-8