N. U. Ahmed - Optimal Control of Dynamic Systems Driven by Vector Measures: Theory and Applications

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There are many prominent areas of systems and control theory that include systems governed by linear and nonlinear ordinary and functional differential equations, stochastic differential equations, partial differential equations including their stochastic counterparts, and, above all, systems governed by abstract differential equations and inclusions. The remarkable advance of this field is due to the unprecedented interest, interaction, and contribution of pure and applied mathematics and physical and engineering sciences. We strongly believe that this interaction will continue simply because there are many unsolved challenging problems and emerging new ones. Such problems are of great interest to mathematicians, scientists, and engineers.

This book is concerned with the development of optimal control theory for finite dimensional systems governed by deterministic as well as stochastic differential equations (which may be subject to impulsive forces) controlled by vector measures. Impulsive controls are special cases of controls determined by vector measures. The book has two major parts. The first part deals with deterministic dynamic systems including differential inclusions controlled by vector measures; the second part is concerned with stochastic dynamic systems also controlled by vector measures. For non-convex control problems, probability measure valued functions known as relaxed controls are used. We consider the question of existence of optimal controls and the necessary conditions of optimality whereby optimal control policies can be determined.

In recent years, significant applications of systems and control theory have been witnessed in areas as diverse as physical sciences, engineering, biological sciences, social sciences, management, and financial engineering, among many others. In particular, the most interesting applications have taken place in areas such as aerospace (civilian, military), space structures (space station, communication satellites), suspension bridges, artificial heart, immunology, power system, hydrodynamics, plasma and magneto hydrodynamics, computer communication networks, and intelligent transportation systems. The importance of applications whereby a theory is tested and new theories are developed is clearly recognized. This book is devoted mainly towards the development of theory of measure-driven differential equations and their optimal control by vector measures.

This book contains seven chapters. In Chap. , we present some practical examples of application with numerical results to demonstrate the applicability of the theories developed in this book.

The authors hope that this book will inspire young mathematicians and mathematically oriented scientists and engineers to further advance the theory and application of dynamic systems driven and controlled by vector measures.

Finally, we would like to appreciate Dr. Remi Lodh, the Editor of Mathematics with Springer Verlag, for his continued support and excellent cooperation throughout the publishing process.