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Volume 2017 (2017), Article ID 5308013, 11 pages
Research Article

Calculus of Variations and Nonlinear Optimization Based Algorithm for Optimal Control of Hybrid Systems with Controlled Switching

1High Institute of Technological Studies in Communications (ISET’COM), Tunis, Tunisia
2Automatic Research Laboratory (LARA), National Engineering School of Tunis (ENIT), Tunis, Tunisia

Correspondence should be addressed to Hajer Bouzaouache

Received 10 February 2017; Revised 7 July 2017; Accepted 10 July 2017; Published 10 August 2017

Academic Editor: Michele Scarpiniti

Copyright © 2017 Hajer Bouzaouache. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper investigates the optimal control problem of a particular class of hybrid dynamical systems with controlled switching. Given a prespecified sequence of active subsystems, the objective is to seek both the continuous control input and the discrete switching instants that minimize a performance index over a finite time horizon. Based on the use of the calculus of variations, necessary conditions for optimality are derived. An efficient algorithm, based on nonlinear optimization techniques and numerical methods, is proposed to solve the boundary-value ordinary differential equations. In the case of linear quadratic problems, the two-point boundary-value problems can be avoided which reduces the computational effort. Illustrative examples are provided and stress the relevance of the proposed nonlinear optimization algorithm.