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

Milan Korda(korda.m***at***gmail.com)
Didier Henrion(henrion***at***laas.fr)
Jean Bernard Lasserre(lasserre***at***laas.fr)

Abstract: 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 \emph{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.

Keywords: Partial differential equations, Occupation measure, Optimal control, Semidefinite programming, Convex optimization

Category 1: Applications -- Science and Engineering (Optimization of Systems modeled by PDEs )

Category 2: Infinite Dimensional Optimization (Distributed Control )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )


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Entry Submitted: 04/19/2018
Entry Accepted: 04/19/2018
Entry Last Modified: 04/19/2018

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