Overlapping Schwarz Scheme for Linear-Quadratic Programs in Continuous Time

TL;DR

The paper develops a continuous-time overlapping Schwarz decomposition for solving time-varying linear-quadratic OCPs by partitioning the horizon into overlapping subintervals and solving local Hamiltonian systems under iteratively updated boundary data. Grounded in Pontryagin’s minimum principle, the method preserves the problem’s continuous structure and supports flexible, potentially adaptive, time integration. A key contribution is proving exponential decay of boundary perturbations (EDS) in the continuous-time setting and deriving explicit linear convergence rates that improve with overlap size. Numerical experiments validate linear convergence, demonstrate benefits of larger overlaps, and showcase robustness with higher-order and adaptive time-stepping solvers, highlighting the approach’s practicality for stiff, large-scale OCPs.

Abstract

We present an optimize-then-discretize framework for solving linear-quadratic optimal control problems (OCP) governed by time-inhomogeneous ordinary differential equations (ODEs). Our method employs a modified overlapping Schwarz decomposition based on the Pontryagin Minimum Principle, partitioning the temporal domain into overlapping intervals and independently solving Hamiltonian systems in continuous time. We demonstrate that the convergence is ensured by appropriately updating the boundary conditions of the individual Hamiltonian dynamics. The cornerstone of our analysis is to prove that the exponential decay of sensitivity (EDS) exhibited in discrete-time OCPs carries over to the continuous-time setting. Unlike the discretize-then-optimize approach, our method can flexibly incorporate different numerical integration methods for solving the resulting Hamiltonian two-point boundary-value subproblems, including adaptive-time integrators. A numerical experiment on a linear-quadratic OCP illustrates the practicality of our approach in broad scientific applications.

Figures (2)

• Figure 1: (Left) Convergence of the overlapping Schwarz method under varying overlap sizes ($\{1\%, 5\%, 10\%, 20\%, 30\%, 60\%\}$ of the subdomain length). Larger overlaps yield faster convergence. (Right) Observed convergence rate and its exponential fit. For each overlap size, we estimate the convergence rate by averaging the ratios of two consecutive errors, and fit an exponential curve of the theoretical form $c\exp(-\rho_Z \tau)$. The fitted values are $\hat{c} = 0.98$ and $\hat{\rho}_Z = 0.05$; and we have $\rho_Z<0.02$ at the significance level of 5%. We observe that the convergence rate improves exponentially in the overlap size/proportion, which is consistent with \ref{['proposition:linear-convergence-in-l-infty']}.
• Figure 2: (Left) Comparison of numerical solvers in resolving the subproblem Hamiltonian dynamics. The Runge-Kutta method (RK45) yields the highest accuracy per iteration. (Right) Comparison of first 30 Schwarz iterations for different solvers and time step sizes $\Delta t$. Adaptive solvers maintain low errors even on coarse grids, while FE accuracy degrades significantly in the stiff regime.

Theorems & Definitions (19)

• Definition 1: Complete Controllability
• Lemma 1: Controllability Gramian
• Remark 1
• Theorem 1: Pontryagin's Minimum Principle (PMP)
• Theorem 2: Hamilton-Jacobi-Bellman (HJB) Equation
• Definition 2
• Proposition 1
• Theorem 3
• Lemma 2: Shifted OCP
• Lemma 3