%0 Journal Article
%T On a Variational Approach to Optimization of Hybrid Mechanical Systems
%A Vadim Azhmyakov
%A Ruben Velazquez
%J Mathematical Problems in Engineering
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/978736
%X This paper deals with multiobjective optimization techniques for a class of hybrid optimal control problems in mechanical systems. We deal with general nonlinear hybrid control systems described by boundary-value problems associated with hybrid-type Euler-Lagrange or Hamilton equations. The variational structure of the corresponding solutions makes it possible to reduce the original ※mechanical§ problem to an auxiliary multiobjective programming reformulation. This approach motivates possible applications of theoretical and computational results from multiobjective optimization related to the original dynamical optimization problem. We consider first order optimality conditions for optimal control problems governed by hybrid mechanical systems and also discuss some conceptual algorithms. 1. Introduction Hybrid and switched systems have been extensively studied in the past decade, both in theory and practice [1每10]. In particular, driven by engineering requirements, there has been increasing interest in optimal control (OC) of these dynamical systems [1每3, 6, 8, 9, 11每14]. In this paper, we investigate some specific types of hybrid systems, namely hybrid systems of mechanical nature, and the corresponding hybrid optimal control problems. The class of problems to be discussed in this work concerns hybrid systems where discrete transitions are being triggered by the continuous dynamics. The control objective is to minimize a cost functional, where the control parameters are usual control inputs. Recently, there has been considerable effort to develop theoretical and computational frameworks for complex control problems. Of particular importance is the ability to operate such systems in an optimal manner. In many real-world applications a controlled mechanical system presents the main modelling framework and is a strongly nonlinear dynamical system of high order [15每17]. Moreover, the majority of applied optimal control problems governed by sophisticated mechanical systems are problems of hybrid nature. The most real-world mechanical control problems are becoming too complex to allow analytical solution. Thus, computational algorithms are inevitable in solving these problems. There is a number of results scattered in the literature on numerical methods for optimal control problems. One can find a fairly complete review in [1, 2, 11, 12, 18每20]. The aim of our investigations is to use the variational structure of the solution to the two-point boundary-value problem for the controllable hybrid-type Euler-Lagrange or Hamilton equation and to propose a new
%U http://www.hindawi.com/journals/mpe/2010/978736/