Optimal Control

by Vinter, R. B.

• ISBN13: 9780817640750
• ISBN10: 0817640754
• Format: Hardcover
• Copyright: 2000-04-01
• Publisher: Birkhauser

Summary

Optimal control emerged as a distinct field of research only in recent decades. It provides a unified perspective of optimization problems, arising in scheduling and the control of engineering devices, that are beyond the reach of traditional analytical and computational techniques. In addition, the field has contributed significant advances to branches of applied mathematics and broad applications in process control, scheduling, robotics, resource economics, and other areas. Optimal Control brings together many of the important advances in 'nonsmooth' optimal control over the last two decades concerning necessary conditions, minimizer regularity and global optimality conditions associated with the Hamilton-Jacobi equation. The book's development and analysis is largely self-contained and incorporates numerous simplifications and unifying features for the subjects' key concepts and foundations. Features and Topics: * a comprehensive overview is provided for specialists and nonspecialists * authoritative, coherent and accessible coverage of the role of nonsmooth analysis in investigating minimizing curves for optimal control * chapter coverage of dynamic programming*explains the necessary conditions for nonconvex problems * chapter coverage of the regularity of minimizers This new book is an essential resource for an authoritative and comprehensive presentation of the foundations and application of nonsmooth optimal control. Postgraduates, researchers and professionals in systems, control, optimization and applied mathematics will find that the book is accessible, provides a rich source of insights and offers clear exposition.

Table of Contents

Preface	xi
Notation	xvii
Overview	1	(60)
Optimal Control	1	(3)
The Calculus of Variations	4	(14)
Existence of Minimizers and Tonelli's Direct Method	18	(3)
Sufficient Conditions and the Hamilton-Jacobi Equation	21	(4)
The Maximum Principle	25	(5)
Dynamic Programming	30	(5)
Nonsmoothness	35	(4)
Nonsmooth Analysis	39	(13)
Nonsmooth Optimal Control	52	(4)
Epilogue	56	(4)
Notes for Chapter 1	60	(1)
Measurable Multifunctions and Differential Inclusions	61	(48)
Introduction	61	(1)
Convergence of Sets	62	(2)
Measurable Multifunctions	64	(11)
Existence and Estimation of F-Trajectories	75	(11)
Perturbed Differential Inclusions	86	(5)
Existence of Minimizing F-Trajectories	91	(3)
Relaxation	94	(6)
The Generalized Bolza Problem	100	(8)
Notes for Chapter 2	108	(1)
Variational Principles	109	(18)
Introduction	109	(1)
Exact Penalization	110	(1)
Ekeland's Theorem	111	(4)
Mini-Max Theorems	115	(10)
Notes for Chapter 3	125	(2)
Nonsmooth Analysis	127	(52)
Introduction	127	(1)
Normal Cones	128	(5)
Subdifferentials	133	(6)
Difference Quotient Representations	139	(5)
Nonsmooth Mean Value Inequalities	144	(5)
Characterization of Limiting Subgradients	149	(5)
Subgradients of Lipschitz Continuous Functions	154	(7)
The Distance Function	161	(5)
Criteria for Lipschitz Continuity	166	(4)
Relationships Between Normal and Tangent Cones	170	(9)
Subdifferential Calculus	179	(22)
Introduction	179	(2)
A Marginal Function Principle	181	(4)
Partial Limiting Subgradients	185	(2)
A Sum Rule	187	(3)
A Nonsmooth Chain Rule	190	(3)
Lagrange Multiplier Rules	193	(4)
Notes for Chapters 4 and 5	197	(4)
The Maximum Principle	201	(32)
Introduction	201	(2)
The Maximum Principle	203	(5)
Derivation of the Maximum Principle from the Extended Euler Condition	208	(6)
A Smooth Maximum Principle	214	(14)
Notes for Chapter 6	228	(5)
The Extended Euler-Lagrange and Hamilton Conditions	233	(52)
Introduction	233	(4)
Properties of the Distance Function	237	(5)
Necessary Conditions for a Finite Lagrangian Problem	242	(10)
The Extended Euler-Lagrange Condition: Nonconvex Velocity Sets	252	(7)
The Extended Euler-Lagrange Condition: Convex Velocity Sets	259	(5)
Dualization of the Extended Euler-Lagrange Condition	264	(13)
The Extended Hamilton Condition	277	(3)
Notes for Chapter 7	280	(5)
Necessary Conditions for Free End-Time Problems	285	(36)
Introduction	285	(3)
Lipschitz Time Dependence	288	(7)
Essential Values	295	(2)
Measurable Time Dependence	297	(4)
Proof of Theorem 8.4.1	301	(9)
Proof of Theorem 8.4.2	310	(3)
A Free End-Time Maximum Principle	313	(5)
Notes for Chapter 8	318	(3)
The Maximum Principle for State Constrained Problems	321	(40)
Introduction	321	(3)
Convergence of Measures	324	(5)
The Maximum Principle for Problems with State Constraints	329	(5)
Derivation of the Maximum Principle for State Constrained Problems from the Euler-Lagrange Condition	334	(5)
A Smooth Maximum Principle for State Constrained Problems	339	(20)
Notes for Chapter 9	359	(2)
Differential Inclusions with State Constraints	361	(36)
Introduction	361	(1)
A Finite Lagrangian Problem	362	(6)
The Extended Euler-Lagrange Condition for State Constrained Problems: Nonconvex Velocity Sets	368	(7)
Necessary Conditions for State Constrained Problems: Convex Velocity Sets	375	(7)
Free Time Problems with State Constraints	382	(5)
Nondegenerate Necessary Conditions	387	(9)
Notes for Chapter 10	396	(1)
Regularity of Minimizers	397	(38)
Introduction	397	(6)
Tonelli Regularity	403	(5)
Proof of The Generalized Tonelli Regularity Theorem	408	(9)
Lipschitz Continuous Minimizers	417	(5)
Autonomous Variational Problems with State Constraints	422	(3)
Bounded Controls	425	(3)
Lipschitz Continuous Controls	428	(4)
Notes for Chapter 11	432	(3)
Dynamic Programming	435	(58)
Introduction	435	(7)
Invariance Theorems	442	(10)
The Value Function and Generalized Solutions of the Hamilton-Jacobi Equation	452	(13)
Local Verification Theorems	465	(9)
Adjoint Arcs and Gradients of the Value Function	474	(9)
State Constrained Problems	483	(4)
Notes for Chapter 12	487	(6)
References	493	(12)
Index	505