[Paper Review] Stability and Optimal Control of Switching PDE-Dynamical Systems

Falk M. Hante|arXiv (Cornell University)|Feb 22, 2018

Stability and Controllability of Differential Equations5 references3 citations

TL;DR

This paper develops a theoretical framework for stability and optimal control of switching systems governed by partial differential equations (PDEs), using abstract evolution equations in Banach and Hilbert spaces. It establishes conditions for uniform exponential stability via common Lyapunov functions, derives generalized observability inequalities for intermittently damped systems, and provides a relaxation-based method for optimal switching control with verified convergence and optimality gaps, demonstrated numerically on PDEs including the heat, Schrödinger, wave, and transport equations.

ABSTRACT

Selected results for the stability and optimal control of abstract switched systems in Banach and Hilbert space are reviewed. The dynamics are typically given in a piecewise sense by a family of nonlinearly perturbed evolutions of strongly continuous semigroups. Stability refers to characterizations of asymptotic decay of solutions that holds uniformly for certain classes of switching signals for time going to infinity. Optimal control refers to the minimization of costs associated to solutions by appropriately selecting switching signals. Selected numerical results verify and visualize some of the available theory.

Motivation & Objective

• To establish sufficient and necessary conditions for uniform exponential stability of switching PDE-dynamical systems under arbitrary switching signals.
• To develop a theory of generalized observability inequalities for asymptotic stability in intermittently damped dissipative systems.
• To analyze the relaxation gap and regularity of the optimal value function in optimal switching control problems.
• To derive an adjoint calculus for perturbations in switching times and mode insertions in optimal control.
• To validate theoretical results through numerical case studies on PDEs including heat, Schrödinger, wave, and transport equations.

Proposed method

• Formulates switching PDE-dynamical systems as abstract evolution equations with piecewise-defined semigroup generators and Lipschitz nonlinear perturbations.
• Applies the theory of strongly continuous semigroups and mild solutions to define solutions over switching intervals.
• Uses common Lyapunov functions to characterize global uniform exponential stability across switching signals.
• Derives generalized observability inequalities to ensure asymptotic stability in systems with intermittent damping.
• Employs outer convexification and relaxation techniques to solve mixed-integer nonlinear programming (MINLP) formulations of optimal control problems.
• Uses finite-volume and implicit Euler schemes for spatial and temporal discretization, with GAMS solvers (BONMIN, BARON) to compute optimal switching sequences.

Experimental results

Research questions

• RQ1 What conditions ensure uniform exponential stability of switching PDE systems under arbitrary switching signals?
• RQ2 How can generalized observability inequalities be used to establish asymptotic stability in intermittently damped systems?
• RQ3 What is the size of the relaxation gap in optimal switching control problems involving switching costs?
• RQ4 How does the regularity of the optimal value function depend on switching time perturbations and mode insertions?
• RQ5 Can relaxation techniques yield exact or ε-optimal solutions for optimal switching control in PDE systems?

Key findings

• For a 1D transport equation test case, the relaxation-based method achieved a relative integer optimality gap of 0% at Δt = 0.25 with Nx = 1000, confirming theoretical convergence rates.
• The MINLP solver GAMS/BONMIN found the optimal switching signal λ(t) = 1 − χ(9/4,T](t) for a 1D heat equation with lumped control, with computation time growing exponentially with Nx.
• The outer convexification/relaxation method achieved a CPU time that grew linearly with Nx, remaining feasible up to Nx = 1000, while MINLP solvers failed beyond Nx = 70.
• For Nx = 1000, the relaxed control solution matched the MINLP solution for Δt ≤ 0.25, validating the relaxation approach.
• The computed optimal cost J* decreased from 0.280 at Nx = 10 to 0.0742 at Nx = 1000, indicating convergence of the numerical scheme.
• The relative optimality gap γ(α³) = 0 for Δt = 0.25 confirmed the theoretical prediction in Theorem 7 that the gap vanishes as Δt → 0.