This book introduces a number of recent advances regarding periodic feedback stabilization for linear and time periodic evolution equations. First, it presents selected connections between linear quadratic optimal control theory and feedback stabilization theory for linear periodic evolution equations. Secondly, it identifies several criteria for the periodic feedback stabilization from the perspective of geometry, algebra and analyses respectively. Next, it describes several ways to design periodic feedback laws. Lastly, the book introduces readers to key methods for designing the control machines. Given its coverage and scope, it offers a helpful guide for graduate students and researchers in the areas of control theory and applied mathematics.
Prof. Gengsheng Wang received his Ph.D. in Mathematics from Ohio University (Athens, USA) in 1994. He is currently Luojia Professor at the School of Mathematics and Statistics, and Director of the Institute of Mathematics, at Wuhan University, China. His research work mainly focus on optimal control problems, (in particular, time optimal control problems), controllability and stabilization for ordinary differential equations, and partial differential equations of parabolic type.
Dr. Yashan Xu received his Ph.D. in Mathematics from Fudan University (Shanghai, China) in 2006. He is currently an Associate Professor at the School of Mathematical Sciences, Fudan University, China. His research interests include differential games, optimal control theory and stabilization for evolution equations.
1 Controlled Periodic Equations, LQ Problems and Periodic Stabilization. 1.1 Controlled Periodic Evolution Equations. 1.2 Linear Quadratic Optimal Control Problems. 1.2.1 Finite horizon case. 1.2.2 Infinite horizon case. 1.3 Relation between Periodic Stabilization and LQ Problems. 2 Criteria on Periodic Stabilization in Infinite Dimensional Cases. 2.1 Attainable Subspaces. 2.2 Three Criterions on Periodic Feedback Stabilization. 2.2.1 Multi-periodic feedback stabilization. 2.2.2 Proof of Theorem 2.1. 2.3 Applications2.3.1 Feedback realization in finite dimensional subspaces. 2.3.2 Applications to heat equations. 3 Criteria on Periodic Stabilization in Finite Dimensional Cases. 3.1 Null Controllable Subspaces. 3.2 Algebraic Criterion and Application. 3.2.1 The proof of (a),(c) in Theorem 3.1. 3.2.2 The proof of (a),(b) in Theorem 3.1. 3.2.3 Decay rate of stabilized equations. 3.3 Geometric Criterion. 4 Design of Simple Control Machines. 4.1 The First Kind of Simple Control Machines. 4.2 The Second Kind of Simple Control Machines-General Case. 4.3 The Second Kind of Simple Control Machines-Special Case