[Paper Review] Guaranteed optimal reachability control of reaction-diffusion equations using one-sided Lipschitz constants and model reduction

Adrien Le Coënt, Laurent Fribourg|arXiv (Cornell University)|Jul 25, 2019

Model Reduction and Neural Networks39 references4 citations

TL;DR

This paper presents a guaranteed optimal control method for 1D reaction-diffusion equations using explicit Euler time discretization and one-sided Lipschitz (OSL) constants, proving convergence and linear error bounds in horizon length when the OSL constant is negative. It further introduces a model reduction technique to improve computational efficiency for larger systems.

ABSTRACT

We show that, for any spatially discretized system of reaction-diffusion, the approximate solution given by the explicit Euler time-discretization scheme converges to the exact time-continuous solution, provided that diffusion coefficient be sufficiently large. By "sufficiently large", we mean that the diffusion coefficient value makes the one-sided Lipschitz constant of the reaction-diffusion system negative. We apply this result to solve a finite horizon control problem for a 1D reaction-diffusion example. We also explain how to perform model reduction in order to improve the efficiency of the method.

Motivation & Objective

• Address the challenge of finite horizon optimal control for nonlinear reaction-diffusion PDEs with guaranteed reachability.
• Ensure numerical stability and convergence of time-discretized solutions by leveraging negative one-sided Lipschitz constants.
• Develop a computationally efficient method by combining explicit Euler integration with model order reduction.
• Provide rigorous error bounds that scale linearly with time horizon, improving upon exponential bounds in prior work.
• Enable application of optimal control to larger spatial discretizations through projection-based model reduction.

Proposed method

• Apply spatial semi-discretization to convert the reaction-diffusion PDE into a system of ODEs.
• Use explicit Euler time integration for numerical solution, with convergence guaranteed when the OSL constant of the system is negative.
• Leverage the one-sided Lipschitz (OSL) constant to derive a priori error bounds that are linear in the time horizon T.
• Apply proper orthogonal decomposition (POD) for model reduction, projecting high-dimensional systems onto lower-dimensional subspaces.
• Construct a reduced system where the optimal control strategy is synthesized, then applied to the full system with bounded error.
• Use projection-based error estimation to bound the deviation between full and reduced system trajectories, with error proportional to $ \frac{K_2 \sigma}{|\lambda_{h_1}|} $.

Experimental results

Research questions

• RQ1 Can explicit Euler time discretization be guaranteed to converge to the true solution of a reaction-diffusion PDE under specific conditions?
• RQ2 What is the impact of the one-sided Lipschitz constant on error bounds in finite-horizon optimal control of PDEs?
• RQ3 How can model reduction be applied to reaction-diffusion systems to maintain control performance while reducing computational cost?
• RQ4 Can the error bound between reduced and full system trajectories be quantitatively bounded in a way that supports guaranteed control synthesis?
• RQ5 Does the proposed method yield linear error scaling with time horizon, in contrast to exponential scaling in prior approaches?

Key findings

• The explicit Euler scheme converges to the exact solution of the reaction-diffusion PDE when the diffusion coefficient is sufficiently large to ensure a negative one-sided Lipschitz constant.
• The error bound for the optimal control solution scales linearly with the time horizon T, in contrast to the exponential dependence $ O(e^{L_f T}) $ found in prior work.
• Model reduction via projection reduces system size significantly—e.g., from $ M_2 = 10 $ to $ M_1 = 5 $—reducing grid size by a factor of approximately $ 15^5 \approx 7.6 \times 10^5 $, making large-scale control feasible.
• The reduction error is bounded by $ \frac{K_2 \sigma}{|\lambda_{h_1}|} $, with $ \sigma $ representing the POD truncation error, and this bound ensures that the projected trajectory remains within a known neighborhood of the target.
• Empirical results for a 1D bi-stable reaction-diffusion system show that the control strategy derived from the reduced model successfully steers the full system toward the target state, with simulation results closely matching those of the full system.
• A posteriori simulations show that the theoretical error bound is conservative, with actual distances between trajectories being significantly smaller than the upper bound.