Extremum and Nash Equilibrium Seeking with Delays and PDEs: Designs & Applications

TL;DR

This paper reviews advanced extremum seeking and Nash equilibrium algorithms for infinite-dimensional systems described by PDEs and delays, highlighting new designs, theoretical insights, and diverse engineering applications.

Contribution

It introduces and analyzes extremum seeking and Nash equilibrium seeking methods for PDE-based systems with delays, expanding the scope of adaptive control to infinite-dimensional dynamics.

Findings

Developed ES algorithms for hyperbolic and parabolic PDEs with delays.

Extended NES methods to heterogeneous PDE game scenarios.

Illustrated applications in urban mobility, oil drilling, and biological systems.

Abstract

The development of extremum seeking (ES) has progressed, over the past hundred years, from static maps, to finite-dimensional dynamic systems, to networks of static and dynamic agents. Extensions from ODE dynamics to maps and agents that incorporate delays or even partial differential equations (PDEs) is the next natural step in that progression through ascending research challenges. This paper reviews results on algorithm design and theory of ES for such infinite-dimensional systems. Both hyperbolic and parabolic dynamics are presented: delays or transport equations, heat-dominated equation, wave equations, and reaction-advection-diffusion equations. Nash equilibrium seeking (NES) methods are introduced for noncooperative game scenarios of the model-free kind and then specialized to single-agent optimization. Even heterogeneous PDE games, such as a duopoly with one parabolic and one hyperbolic agent, are considered. Several engineering applications are touched upon for illustration, including flow-traffic control for urban mobility, oil-drilling systems, deep-sea cable-actuated source seeking, additive manufacturing modeled by the Stefan PDE, biological reactors, light-source seeking with flexible-beam structures, and neuromuscular electrical stimulation.