Infinite-Dimensional Sums-of-Squares for Optimal Control

TL;DR

This paper presents an innovative approach to solving optimal control problems using an infinite-dimensional sum-of-squares representation within a reproducing kernel Hilbert space, enabling practical semi-definite programming solutions.

Contribution

It extends sum-of-squares methods from polynomial to smooth problems using reproducing kernel Hilbert spaces, providing a new approximation technique for optimal control.

Findings

Successfully applied to a low-dimensional control problem

Demonstrates the feasibility of infinite-dimensional sum-of-squares representations

Provides a practical semi-definite programming implementation

Abstract

We introduce an approximation method to solve an optimal control problem via the Lagrange dual of its weak formulation. It is based on a sum-of-squares representation of the Hamiltonian, and extends a previous method from polynomial optimization to the generic case of smooth problems. Such a representation is infinite-dimensional and relies on a particular space of functions-a reproducing kernel Hilbert space-chosen to fit the structure of the control problem. After subsampling, it leads to a practical method that amounts to solving a semi-definite program. We illustrate our approach by a numerical application on a simple low-dimensional control problem.