Peak Estimation of Time Delay Systems using Occupation Measures

TL;DR

This paper introduces a measure-based approach using occupation measures and semidefinite programming to estimate the maximum of a state function along trajectories of delay differential equations, demonstrated with example applications.

Contribution

It develops a novel measure-based framework for peak estimation in DDEs, extending existing methods to systems with delays using occupation measures and SOS hierarchies.

Findings

Effective peak estimates for DDEs demonstrated

Method applicable to epidemic models with delays

Convergence of SDP hierarchy shown

Abstract

This work proposes a method to compute the maximum value obtained by a state function along trajectories of a Delay Differential Equation (DDE). An example of this task is finding the maximum number of infected people in an epidemic model with a nonzero incubation period. The variables of this peak estimation problem include the stopping time and the original history (restricted to a class of admissible histories). The original nonconvex DDE peak estimation problem is approximated by an infinite-dimensional Linear Program (LP) in occupation measures, inspired by existing measure-based methods in peak estimation and optimal control. This LP is approximated from above by a sequence of Semidefinite Programs (SDPs) through the moment-Sum of Squares (SOS) hierarchy. Effectiveness of this scheme in providing peak estimates for DDEs is demonstrated with provided examples