Nonlinear and Optimal Control Theory: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, June 19-29, 2004

ISBN-13: 9783540776444 / Angielski / Miękka / 2008 / 360 str.

Andrei Agrachev; A. Stephen Morse; Eduardo D. Sontag

Opis

Mathematical Control Theory is a branch of Mathematics having as one of its main aims the establishment of a sound mathematical foundation for the c- trol techniques employed in several di?erent ?elds of applications, including engineering, economy, biologyandsoforth. Thesystemsarisingfromthese- plied Sciences are modeled using di?erent types of mathematical formalism, primarily involving Ordinary Di?erential Equations, or Partial Di?erential Equations or Functional Di?erential Equations. These equations depend on oneormoreparameters thatcanbevaried, andthusconstitute thecontrol - pect of the problem. The parameters are to be chosen soas to obtain a desired behavior for the system. From the many di?erent problems arising in Control Theory, the C. I. M. E. school focused on some aspects of the control and op- mization ofnonlinear, notnecessarilysmooth, dynamical systems. Two points of view were presented: Geometric Control Theory and Nonlinear Control Theory. The C. I. M. E. session was arranged in ?ve six-hours courses delivered by Professors A. A. Agrachev (SISSA-ISAS, Trieste and Steklov Mathematical Institute, Moscow), A. S. Morse (Yale University, USA), E. D. Sontag (Rutgers University, NJ, USA), H. J. Sussmann (Rutgers University, NJ, USA) and V. I. Utkin (Ohio State University Columbus, OH, USA). We now brie?y describe the presentations. Agrachev s contribution began with the investigation of second order - formation in smooth optimal control problems as a means of explaining the variational and dynamical nature of powerful concepts and results such as Jacobi ?elds, Morse s index formula, Levi-Civita connection, Riemannian c- vature."

Spis treści

Geometry of Optimal Control Problems and Hamiltonian Systems.- Lecture Notes on Logically Switched Dynamical Systems.- Input to State Stability: Basic Concepts and Results.- Generalized Differentials, Variational Generators, and the Maximum Principle with State Constraints.- Sliding Mode Control: Mathematical Tools, Design and Applications.

Blurb

The lectures gathered in this volume present some of the different aspects of Mathematical Control Theory. Adopting the point of view of Geometric Control Theory and of Nonlinear Control Theory, the lectures focus on some aspects of the Optimization and Control of nonlinear, not necessarily smooth, dynamical systems. Specifically, three of the five lectures discuss respectively: logic-based switching control, sliding mode control and the input to the state stability paradigm for the control and stability of nonlinear systems. The remaining two lectures are devoted to Optimal Control: one investigates the connections between Optimal Control Theory, Dynamical Systems and Differential Geometry, while the second presents a very general version, in a non-smooth context, of the Pontryagin Maximum Principle. The arguments of the whole volume are self-contained and are directed to everyone working in Control Theory. They offer a sound presentation of the methods employed in the control and optimization of nonlinear dynamical systems.