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Chapitre D'ouvrage Année : 2022

Moments and convex optimization for analysis and control of nonlinear partial differential equations

Résumé

This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a linear equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is then approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems (SDPs). In the case of analysis of uncontrolled PDEs, the solutions to these SDPs provide bounds on a specified, possibly nonlinear, functional of the solutions to the PDE; in the case of PDE control, the solutions to these SDPs provide bounds on the optimal value of a given optimal control problem as well as suboptimal feedback controllers. The entire approach is based purely on convex optimization and does not rely on spatio-temporal gridding, even though the PDE addressed can be fully nonlinear. The approach is applicable to a very broad class nonlinear PDEs with polynomial data. Computational complexity is analyzed and several complexity reduction procedures are described. Numerical examples demonstrate the approach.
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Dates et versions

hal-01771699 , version 1 (20-04-2018)

Identifiants

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Milan Korda, Didier Henrion, Jean-Bernard Lasserre. Moments and convex optimization for analysis and control of nonlinear partial differential equations. Elsevier. Handbook of Numerical Analysis, 23, Elsevier, pp.339--366, 2022. ⟨hal-01771699⟩
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