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Abstract
This paper presents a numerical scheme for solving the continuous-time convex linear-quadratic (LQ) optimal control problem with mixed polyhedral state and control constraints. Unifying a discretization of this optimal control problem as often employed in model predictive control and that obtained through time-stepping methods based on the differential variational inequality reformulation, the scheme solves a sequence of finite-dimensional convex quadratic programs (QPs) whose optimal solutions are employed to construct a sequence of discrete-time trajectories dependent on the time step. Under certain technical primal–dual assumptions primarily to deal with the algebraic constraints involving the state variable, we prove that such a numerical trajectory converges to an optimal trajectory of the continuous-time control problem as the time step goes to zero, with both the limiting optimal state and costate trajectories being absolutely continuous. This provides a constructive proof of the existence of a solution to the optimal control problem with such regularity properties. Additional properties of the optimal solutions to the LQ problem are also established that are analogous to those of the finite-dimensional convex QP. Our results are applicable to problems with convex but not necessarily strictly convex objective functions and with possibly unbounded mixed state–control constraints.
Acknowledgements
This paper has gone through an extensive review and has thus benefitted from many constructive comments of the referees including suggestions of related references. The authors thank all the referees. We acknowledge discussion with Dr Rafal Goebel in the early stage of the paper regarding the state-constrained optimal control problem and thank him for calling our attention to the penalty approach of Rockafellar and related references. We also thank Dane Schiro for pointing out some typos. The work of the first author was based on research partially supported by the National Science Foundation under grants DMS-0754374 and CMMI-0969600. The fourth author was partially supported by the European Commission through project MOBY-DIC ‘Model-based synthesis of digital electronic circuits for embedded control’ (FP7-INFSO-ICT-248858), http://www.mobydic-project.eu.
Notes
†The authors are very pleased to dedicate this paper to Professor Florian A. Potra on the occasion of his 60th birthday. The topic of this paper lies at the intersection of several of Professor Potra's areas of expertise: complementarity problems, differential-algebraic systems, numerical optimization, optimal control of mechanical systems, in each of which he has made fundamental contributions. In particular, the papers Citation2 Citation3 Citation36 Citation61 by Professor Potra, the first guest editor of this special issue, Dr Mihai Anitescu, address the convergence of linear complementarity based time-stepping methods for solving multi-body contact problems with friction, whose differential-algebraic formulation has provided an important motivation for the introduction of the class of differential variational inequalities (DVIs) that has provided the basic framework for the optimal control problem studied herein.