Optimal Impulsive Control : The Extension Approach

__Optimal Impulsive Control__ explores the class of impulsive dynamic optimization problems—problems that stem from the fact that many conventional optimal control problems do not have a solution in the classical setting—which is highly relevant with regard to engineering applications. The absence of a classical solution naturally invokes the so-called extension, or relaxation, of a problem, and leads to the notion of generalized solution which encompasses the notions of generalized control and trajectory; in this book several extensions of optimal control problems are considered within the framework of optimal impulsive control theory. In this framework, the feasible arcs are permitted to have jumps, while the conventional absolutely continuous trajectories may fail to exist. The authors draw together various types of their own results, centered on the necessary conditions of optimality in the form of Pontryagin’s maximum principle and the existence theorems, which shape a substantial body of optimal impulsive control theory. At the same time, they present optimal impulsive control theory in a unified framework, introducing the different paradigmatic problems in increasing order of complexity. The rationale underlying the book involves addressing extensions increasing in complexity from the simplest case provided by linear control systems and ending with the most general case of a totally nonlinear differential control system with state constraints.The mathematical models presented in Optimal Impulsive Control being encountered in various engineering applications, this book will be of interest to both academic researchers and practising engineers. Preface......Page 6 Contents......Page 8 Notation......Page 10 List of Assertions......Page 12 Introduction......Page 14 References......Page 22 1.1 Introduction......Page 25 1.2 Problem Statement......Page 30 1.3 Existence Theorem......Page 33 1.4 Maximum Principle......Page 35 1.5 Exercises......Page 40 References......Page 42 2.1 Introduction......Page 43 2.2 Problem Statement......Page 44 2.3 Maximum Principle......Page 46 2.4 Proof of Lemma 2.1......Page 49 2.5 Exercises......Page 61 References......Page 62 3.1 Introduction......Page 63 3.2 Problem Statement......Page 64 3.3 Preliminaries......Page 68 3.4 Maximum Principle......Page 76 3.5 Second-Order Optimality Conditions for a Simple Problem......Page 80 3.6 Second-Order Necessary Conditions Under the Frobenius Condition......Page 86 3.7 Exercises......Page 95 References......Page 97 6 Impulsive Control Problems with Mixed Constraints......Page 0 4.1 Introduction......Page 99 4.2 Problem Statement and Solution Concept......Page 101 4.3 Well-Posedness......Page 103 4.4 Existence of Solution......Page 112 4.5 Maximum Principle......Page 114 4.6 Exercises......Page 119 References......Page 121 5.1 Introduction......Page 122 5.2 Problem Statement......Page 124 5.3 Maximum Principle in Gamkrelidze's Form......Page 127 5.4 Nondegeneracy Conditions......Page 137 5.5 Exercises......Page 139 References......Page 140 6.1 Introduction......Page 143 6.2 Example......Page 145 6.3 Problem Formulation and Basic Definitions......Page 148 6.4 Basic Constructions and Lemmas......Page 152 6.5 Maximum Principle......Page 160 6.6 Exercises......Page 172 References......Page 173 7.1 Introduction......Page 175 7.2 Preliminaries......Page 178 7.3 Extension Concept......Page 181 7.4 Generalized Existence Theorem......Page 183 7.5 Maximum Principle......Page 189 7.6 Examples of Extension......Page 191 7.7 Exercises......Page 193 References......Page 194 Index......Page 195 Optimal Impulsive Control explores the class of impulsive dynamic optimization problems—problems that stem from the fact that many conventional optimal control problems do not have a solution in the classical setting—which is highly relevant with regard to engineering applications. The absence of a classical solution naturally invokes the so-called extension, or relaxation, of a problem, and leads to the notion of generalized solution which encompasses the notions of generalized control and trajectory; in this book several extensions of optimal control problems are considered within the framework of optimal impulsive control theory. In this framework, the feasible arcs are permitted to have jumps, while the conventional absolutely continuous trajectories may fail to exist. The authors draw together various types of their own results, centered on the necessary conditions of optimality in the form of Pontryagin’s maximum principle and the existence theorems, which shape a substantial body of optimal impulsive control theory. At the same time, they present optimal impulsive control theory in a unified framework, introducing the different paradigmatic problems in increasing order of complexity. The rationale underlying the book involves addressing extensions increasing in complexity from the simplest case provided by linear control systems and ending with the most general case of a totally nonlinear differential control system with state constraints. The mathematical models presented in Optimal Impulsive Control being encountered in various engineering applications, this book will be of interest to both academic researchers and practising engineers. "Optimal Impulsive Control explores the class of impulsive dynamic optimization problems--problems that stem from the fact that many conventional optimal control problems do not have a solution in the classical setting--which is highly relevant with regard to engineering applications. The absence of a classical solution naturally invokes the so-called extension, or relaxation, of a problem, and leads to the notion of generalized solution which encompasses the notions of generalized control and trajectory; in this book several extensions of optimal control problems are considered within the framework of optimal impulsive control theory. In this framework, the feasible arcs are permitted to have jumps, while the conventional absolutely continuous trajectories may fail to exist. The authors draw together various types of their own results, centered on the necessary conditions of optimality in the form of Pontrygin's maximum principle and the existence theorems, which shape a substantial body of optimal impulsive control theory. At the same time, they present optimal impulsive control theory in a unified framework, introducing the different paradigmatic problems in increasing order of complexity. The rationale underlying the book involves addressing extensions increasing in complexity from the simplest case provided by linear control systems and ending with the most general case of a totally nonlinear differential control system with state constraints."--Back cover