A Semi-smooth Newton Method for the Constrained Optimal Control of Continuous-Time Linear Systems

TL;DR

This paper introduces a semi-smooth Newton method for solving constrained optimal control problems in continuous time, reformulating KKT conditions as a rootfinding problem and solving via a modified Riccati equation.

Contribution

It presents a novel indirect approach that embeds KKT conditions into a non-smooth complementarity function and solves the resulting rootfinding problem with a semi-smooth Newton method.

Findings

Method converges superlinearly up to ODE solver tolerance.

Reformulation as rootfinding simplifies handling constraints.

Numerical simulations demonstrate effectiveness and convergence.

Abstract

This paper details a novel indirect method for solving constrained optimal control problems (OCPs) directly in continuous-time function space. The KKT conditions are embedded in a non-smooth complementarity function, which enables their reformulation as a rootfinding problem in Banach space. This problem is then solved using a non-smooth Newton method. Finally, the paper shows that the Newton update can be obtained by solving a modified differential Riccati equation, where the cost terms are reweighted at every iteration based on the constraint multipliers. Numerical simulations show the effectiveness of the method, which converges superlinearly up to the tolerance of the ODE solver.