ISSN 0005-1179, Automation and Remote Control, 2009, Vol. 70, No. 5, pp. 787–799. c  Pleiades Publishing, Ltd., 2009. Original Russian Text c  V.V. Azmyakov, R. Galvan-Guerra, A.E. Polyakov, 2009, published in Avtomatika i Telemekhanika, 2009, No. 5, pp. 51–64. DETERMINATE SYSTEMS On the Method of Dynamic Programming for Linear-Quadratic Problems of Optimal Control in Hybrid Systems V. V. Azmyakov, ∗ R. Galvan-Guerra, ∗ and A. E. Polyakov ∗∗ * Research and Advanced Studies Center, National Polytechnical Institute, Mexico, United Mexican States ** Voronezh State University, Voronezh, Russia Received May 21, 2008 Abstract—For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid linear-quadratic optimal control based on the recently suggested principle of maximum. Interrelations between the hybrid principle of maximum and the method of dynamic programming for the systems of this class were discussed. The classical formalism was extended, the corresponding Riccati equations were obtained, and discontinuity of the “hybrid” Riccati matrix was proved. The computational aspects of the established theoretical results were considered. PACS number: 02.30.Yy DOI: 10.1134/S0005117909050075 1. INTRODUCTION Various types of hybrid and switched systems are used in the modern engineering to model complex control system in chemical, bioengineering, and aero-space industries, as well in the systems of industrial electronics [1–6]. Additionally, the hybrid models are encountered in the multiagent distributed control systems and systems for management of social processes ([4] for example). The control theory for a sufficiently long time considered the discrete-continuous dynamic sys- tems and the switched systems. New outlooks for the studies of some important applications of the variable-structure technical systems were opened by the developing theory of the hybrid systems and corresponding numerical algorithms. The problem of optimal control in the hybrid systems is especially complicated because in the general case one faces not only the infinite-dimensional prob- lem of optimization, but also the need for combinatorial enumeration due to the discrete part of the system. In this connection, the majority of the schemes of practical optimization were proposed for particular classes of the hybrid systems. Many of them were based on the recently established optimality conditions, see, for example, [6, 8–20], the rest being related more with the classical approaches to dynamic system optimization [1–3, 21, 22]. Recently, an interest to the first-order optimization methods and the corresponding computational schemes based on the gradient meth- ods and the principle of maximum [8–11] was displayed. This is due primarily to the simplicity of their realization, reliability, and availability of the results enabling one to support their accuracy and convergence. At the same time, the dynamic programming-based approaches are inadequate to the hybrid linear systems and corresponding linear-quadratic problems of optimal control. Along with the numerical procedures based on the Pontryagin principle of maximum, for the classical problems of optimal feedback control, the main means of constructing the optimal tra- jectory are represented by the famous Bellman method of dynamic programming. The traditional 787