Closed-Form Characterization of Constrained Double-Integrator Optimal Control

TL;DR

This paper derives explicit closed-form formulas for optimal control trajectories of double-integrator systems with constraints, classifying all possible arc combinations and simplifying the computation of switching times.

Contribution

It provides a comprehensive classification and explicit formulas for all admissible control profiles, including special cases, for energy-optimal constrained double-integrator control.

Findings

Closed-form expressions for switching times of all admissible profiles.

Classification of all possible arc combinations, including special cases.

Optimal trajectories follow a bang-affine-coast profile when initial conditions violate constraints.

Abstract

We consider the energy-optimal control problem for double-integrator systems subject to state and control constraints, with fixed terminal time and free terminal speed. When the constraints become active, the optimal trajectory consists of a combination of bang, unconstrained, and coast arcs, whose switching instants must be computed explicitly. In this paper, we derive closed-form expressions for the switching times of all admissible profiles, including both constrained and unconstrained arcs, reducing the computation in each case to explicit algebraic equations. In contrast to prior work, we classify all possible combinations of arcs, including special cases, and provide the specific conditions under which each case arises. Furthermore, we prove that when the initial unconstrained trajectory violates both speed and control constraints, the optimal solution follows a predetermined bang-affine-coast profile, enabling direct identification of the optimal trajectory without intermediate feasibility checks.