# A Parallel Decomposition Scheme for Solving Long-Horizon Optimal Control Problems

## TL;DR

This paper introduces a parallel temporal decomposition method for long-horizon optimal control problems, ensuring convergence under specific conditions related to boundary perturbation decay and overlap size, with demonstrated effectiveness on quadratic and non-convex problems.

## Contribution

It proposes a novel parallel decomposition scheme with convergence guarantees based on boundary perturbation decay, applicable to both convex and non-convex problems.

## Key findings

- Scheme converges if boundary perturbations decay asymptotically.
- LQ problems satisfy the decay condition, ensuring convergence.
- Overlap size influences convergence rate and scheme performance.

## Abstract

We present a temporal decomposition scheme for solving long-horizon optimal control problems. In the proposed scheme, the time domain is decomposed into a set of subdomains with partially overlapping regions. Subproblems associated with the subdomains are solved in parallel to obtain local primal-dual trajectories that are assembled to obtain the global trajectories. We provide a sufficient condition that guarantees convergence of the proposed scheme. This condition states that the effect of perturbations on the boundary conditions (i.e., initial state and terminal dual/adjoint variable) should decay asymptotically as one moves away from the boundaries. This condition also reveals that the scheme converges if the size of the overlap is sufficiently large and that the convergence rate improves with the size of the overlap. We prove that linear quadratic problems satisfy the asymptotic decay condition, and we discuss numerical strategies to determine if the condition holds in more general cases. We draw upon a non-convex optimal control problem to illustrate the performance of the proposed scheme.