On Integral Linear Constraints on Convex Cones

TL;DR

This paper explores integral linear constraints on trajectories within convex cones, establishing conditions for their satisfaction via bounded linear functionals, and connects these results to linear-quadratic control and positive systems theory.

Contribution

It introduces a conic inequality framework for integral constraints and relates it to classical control results like the bounded real lemma and KYP lemma.

Findings

Characterizes integral linear constraints using conic inequalities.

Establishes a link between these constraints and linear functionals in finite-dimensional spaces.

Provides L1-gain and KYP lemma analogues for positive systems.

Abstract

In this paper, we consider integral linear constraints and the dissipation inequality with linear supply rates for certain sets of trajectories confined pointwise in time to a convex cone which belongs to a finite-dimensional normed vector space. Such constraints are then shown to be satisfied if and only if a bounded linear functional exists which satisfies a conic inequality. This is analogous to the typical situation in which a quadratic supply rate over the entire space is related to a linear matrix inequality. A connection is subsequently drawn precisely to linear-quadratic control: by proper choice of cone, the main results can be applied to produce a known L1-gain analogue to the bounded real lemma in positive systems theory, as well as a non-strict version of the Kalman-Yakubovich-Popov Lemma in linear-quadratic control.