This book focuses on how to implement optimal control problems via the variational method. It studies how to implement the extrema of functional by applying the variational method and covers the extrema of functional with different boundary conditions, involving multiple functions and with certain constraints etc. It gives the necessary and sufficient condition for the (continuous-time) optimal control solution via the variational method, solves the optimal control problems with different boundary conditions, analyzes the linear quadratic regulator & tracking problems respectively in detail, and provides the solution of optimal control problems with state constraints by applying the Pontryagin's minimum principle which is developed based upon the calculus of variations. And the developed results are applied to implement several classes of popular optimal control problems and say minimum-time, minimum-fuel and minimum-energy problems and so on.

As another key branch of optimal control methods, it also presents how to solve the optimal control problems via dynamic programming and discusses the relationship between the variational method and dynamic programming for comparison.

Concerning the system involving individual agents, it is also worth to study how to implement the decentralized solution for the underlying optimal control problems in the framework of differential games. The equilibrium is implemented by applying both Pontryagin's minimum principle and dynamic programming.

The book also analyzes the discrete-time version for all the above materials as well since the discrete-time optimal control problems are very popular in many fields.

Acerca del autor

Prof. Zhongjing Ma received the B.Eng. degree in the field of Automatic Control from Nankai University, Tianjin, China, in 1997, and the M.Eng. and Ph.D. degrees from McGill University, Montreal, QC, Canada, in 2005 and 2009, respectively, advised by Prof. Peter Caines and Prof. Roland Malhame. After a period of time, from January 2009 to September 2010, as a postdoctoral research fellow with Prof. Duncan Callaway and Prof. Ian Hiskens, at the University of Michigan, Ann Arbor, he joined the School of Automation at the University of Beijing Institute of Technology, Beijing, China, in September 2010, as Associate Professor. He is currently Director of Institute of Electrical Engineering. He is an IEEE senior member. His research interests lie in the areas of optimal control, optimization, auction mechanism design, game theory, decentralized optimization of large-scale systems, and their applications in the electrical power systems. He has published more than 50 technical articles in IEEE Trans. on Automatic Control, Automatica, IEEE Trans. on Control Systems Technology, IEEE Trans. on Systems, Man, and Cybernetics: Systems, Control Engineering Practice, IET Generation, Transmission & Distribution etc.

He has taught the one-semester graduate course of Optimal and Robust Control, for 9 years at Beijing Institute of Technology, which has been taken in English. He has published a monograph entitled as “Decentralized Charging Coordination of Large-scale Plug-in Electric Vehicles in Power Systems” in Springer Nature in March 2019. He has joined the editorial board of the journal of “Nonlinear Analysis: Hybrid Systems” of the Elsevier press from February 2018 and has served as Associate Editor since then.

Prof. Suli Zou received the B.S. degree in Electrical Engineering and its Automation and the Ph.D. degree in Control Theory and Control Engineering from the Beijing Institute of Technology, Beijing, China, in 2011 and 2017,respectively. She joined the school of Beijing Institute of Technology as Associate Professor in August 2019. Before that, she was a research fellow in Automatic Control Laboratory, Department of Information Technology and Electrical Engineering, ETH Z�rich, Z�rich, Switzerland.

Her current research interests include decentralized optimization, game theory and auction mechanism design, with particular applications to smart grids, demand response and charging coordination of electric vehicles, dynamic and stochastic game, distributed optimization in distribution systems with high penetration of renewables, and learning methods for grid power management. She has published more than 30 technical articles in IEEE Trans. on Automatic Control, Automatica, IEEE Trans. on Control Systems Technology, IEEE Trans. on Systems, Man, and Cybernetics: Systems, Control Engineering Practice etc.

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