Conditions for strict dissipativity of infinite-dimensional generalized linear-quadratic problems

TL;DR

This paper establishes conditions under which infinite-dimensional linear-quadratic optimal control problems exhibit strict dissipativity, linking it to ellipticity of a Lyapunov operator and spectral properties, with an example involving heat equations.

Contribution

It provides new sufficient conditions for strict dissipativity in infinite-dimensional LQ problems, connecting it to ellipticity and spectral assumptions, and illustrates with a heat equation example.

Findings

Strict dissipativity is equivalent to ellipticity of a Lyapunov-like operator.

Under spectral and orthogonality assumptions, ellipticity holds with detectability.

The results are demonstrated through a heat equation example.

Abstract

We derive sufficient conditions for strict dissipativity for optimal control of linear evolution equations on Hilbert spaces with a cost functional including linear and quadratic terms. We show that strict dissipativity with a particular storage function is equivalent to ellipticity of a Lyapunov-like operator. Further we prove under a spectral decomposition assumption of the underlying generator and an orthogonality condition of the resulting subspaces that this ellipticity property holds under a detectability assumption. We illustrate our result by means of an example involving a heat equation on a one-dimensional domain.