Method for solving a linear-quadratic optimal control problem with intermediate phase constraints
Year: 2026
Type: article
Abstract: This paper presents a constructive numerical method for solving a class of linear-quadratic (LQ) optimal control problems with multivariate controls and intermediate phase constraints. Based on the support method of Gabasov and Kirillova, the proposed algorithm employs a primal-dual framework that operates directly on the continuous-time system, thus avoiding its discretization. The method is an iterative process that generates a sequence of improving admissible support controls. A support control is a pair consisting of a feasible control and a set of active constraints (the support) that defines a local, simplified model of the problem. The iterative improvement is achieved by alternating between a control transformation (primal step) to refine the control for a fixed support, and a support transformation (dual step) to correct the active set. Once the main iterative process converges to an approximate optimal support, a finishing procedure is initiated. This final procedure reformulates the problem as a system of nonlinear equations to compute the precise values of the optimal switching times and potentials. At each iteration, the suboptimality estimate provides a rigorous upper bound on the gap to the optimum, so the algorithm can deliver an $ \varepsilon $-optimal solution for any user-chosen tolerance $ \varepsilon > 0 $. The method is illustrated on a two-vehicle convoy problem, and a scaling study shows it to be faster and more accurate than a direct transcription method as the number of intermediate constraints grows.
DOI: 10.3934/naco.2026028
Authors: Mourad Azi, Mohand-Ouamer Bibi, Ali Zaidi
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7204747323