Stabilization, Optimal and Robust Control

Systems governed by nonlinear partial differential equations (PDEs) arise in many spheres of study. The stabilization and control of such systems, which are the focus of this book, are based around game theory. The robust control methods proposed here have the twin aims of compensating for system disturbances in such a way that a cost function achieves its minimum for the worst disturbances and providing the best control for stabilizing fluctuations with a limited control effort.

Stabilization, Optimal and Robust Control develops robust control of infinite-dimensional dynamical systems derived from time-dependent coupled PDEs associated with boundary-value problems. Rigorous analysis takes into account nonlinear system dynamics, evolutionary and coupled PDE behaviour and the selection of function spaces in terms of solvability and model quality.

Mathematical foundations essential for the required analysis are provided so that the book remains accessible to the non-control-specialist. Following chapters giving a general view of convex analysis and optimization and robust and optimal control, problems arising in fluid-mechanical, biological and materials-scientific systems are laid out in detail; specifically:

. mathematical treatment of nonlinear evolution systems (with and without time-varying delays);

. vortex dynamics in superconducting films and solidification of binary alloys;

. large-scale primitive equations in oceanic dynamics;

. heat transfer in biological tissues;

. population dynamics and resource management;

. micropolar fluid and blood motion.

The combination of mathematical fundamentals with applications of current interest will make this book of much interest to researchers and graduate students looking at complex problems in mathematics, physics and biology as well as to control theorists.

Zusammenfassung

Stabilization, Optimal and Robust Control develops robust control of infinite-dimensional dynamical systems derived from time-dependent coupled PDEs associated with boundary-value problems. Rigorous analysis takes into account nonlinear system dynamics, evolutionary and coupled PDE behaviour and the selection of function spaces in terms of solvability and model quality.

Mathematical foundations are provided so that the book remains accessible to the non-control-specialist. Following chapters giving a general view of convex analysis and optimization and robust and optimal control, problems arising in fluid mechanical, biological and materials scientific systems are laid out in detail.

The combination of mathematical fundamentals with application of current interest will make this book of much interest to researchers and graduate students looking at complex problems in mathematics, physics and biology as well as to control theorists.

Inhalt
General Introduction.- General Introduction.- Convex Analysis and Duality Principles.- Convexity and Topology.- A Brief Overview of Sobolev Spaces.- LegendreFenchel Transformation and Duality.- Lagrange Duality Theory.- General Results and Concepts on Robust and Optimal Control Theory for Evolutive Systems.- Studied Systems and General Results.- Optimal Control Problems.- Stabilization and Robust Control Problem.- Remarks on Numerical Techniques.- Applications in the Biological and Physical Sciences: Modeling and Stabilization.- Vortex Dynamics in Superconductors and GinzburgLandau-type Models.- Multi-scale Modeling of Alloy Solidification and Phase-field Model.- Large-scale Ocean in the Climate System.- Heat Transfer Laws on Temperature Distribution in Biological Tissues.- LotkaVolterra-type Systems with Logistic Time-varying Delays.- Other Systems.