Optimization with affine homogeneous quadratic integral inequality constraints

TL;DR

This paper presents a new method for optimizing linear costs under affine homogeneous quadratic integral inequalities, with applications to PDE stability analysis, using SDP-based outer and inner approximations.

Contribution

It introduces a novel approach to approximate feasible sets of quadratic integral inequalities with LMIs, enabling practical optimization for PDE-related problems.

Findings

Outer approximations via LMIs converge to the feasible set.

Inner approximations provide feasible points and upper bounds.

Implementation in open-source software facilitates application.

Abstract

We introduce a new technique to optimize a linear cost function subject to a one-dimensional affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in stability analysis, input-to-state/output analysis, and control of many systems governed by partial differential equations (PDEs), in particular fluid dynamical systems. First, we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that under mild assumptions a convergent, non-decreasing sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can also be computed with SDPs. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our techniques by solving problems arising from the stability analysis of PDEs.