Nonlinear Optimal Control Theory presents a deep, wide-ranging introduction to the mathematical theory of the optimal control of processes governed by ordinary differential equations and certain types of differential equations with memory. Many examples illustrate the mathematical issues that need to be addressed when using optimal control techniques in diverse areas.
Drawing on classroom-tested material from Purdue University and North Carolina State University, the book gives a unified account of bounded state problems governed by ordinary, integrodifferential, and delay systems. It also discusses Hamilton-Jacobi theory. By providing a sufficient and rigorous treatment of finite dimensional control problems, the book equips readers with the foundation to deal with other types of control problems, such as those governed by stochastic differential equations, partial differential equations, and differential games.
Examples of Control Problems Introduction A Problem of Production Planning Chemical Engineering Flight Mechanics Electrical Engineering The Brachistochrone Problem An Optimal Harvesting Problem Vibration of a Nonlinear Beam Formulation of Control Problems Introduction Formulation of Problems Governed by Ordinary Differential Equations Mathematical Formulation Equivalent Formulations Isoperimetric Problems and Parameter Optimization Relationship with the Calculus of Variations Hereditary Problems Relaxed Controls Introduction The Relaxed Problem; Compact Constraints Weak Compactness of Relaxed Controls Filippov's Lemma The Relaxed Problem; Non-Compact Constraints The Chattering Lemma; Approximation to Relaxed Controls Existence Theorems; Compact Constraints Introduction Non-Existence and Non-Uniqueness of Optimal Controls Existence of Relaxed Optimal Controls Existence of Ordinary Optimal Controls Classes of Ordinary Problems Having Solutions Inertial Controllers Systems Linear in the State Variable Existence Theorems; Non Compact Constraints Introduction Properties of Set Valued Maps Facts from Analysis Existence via the Cesari Property Existence without the Cesari Property Compact Constraints Revisited The Maximum Principle and Some of its Applications Introduction A Dynamic Programming Derivation of the Maximum Principle Statement of Maximum Principle An Example Relationship with the Calculus of Variations Systems Linear in the State Variable Linear Systems The Linear Time Optimal Problem Linear Plant-Quadratic Criterion Problem Proof of the Maximum Principle Introduction Penalty Proof of Necessary Conditions in Finite Dimensions The Norm of a Relaxed Control; Compact Constraints Necessary Conditions for an Unconstrained Problem The Îµ-Problem The Îµ-Maximum Principle The Maximum Principle; Compact Constraints Proof of Theorem 6.3.9 Proof of Theorem 6.3.12 Proof of Theorem 6.3.17 and Corollary 6.3.19 Proof of Theorem 6.3.22 Examples Introduction The Rocket Car A Non-Linear Quadratic Example A Linear Problem with Non-Convex Constraints A Relaxed Problem The Brachistochrone Problem Flight Mechanics An Optimal Harvesting Problem Rotating Antenna Example Systems Governed by Integrodifferential Systems Introduction Problem Statement Systems Linear in the State Variable Linear Systems/The Bang-Bang Principle Systems Governed by Integrodifferential Systems Linear Plant Quadratic Cost Criterion A Minimum Principle Hereditary Systems Introduction Problem Statement and Assumptions Minimum Principle Some Linear Systems Linear Plant-Quadratic Cost Infinite Dimensional Setting Bounded State Problems Introduction Statement of the Problem Îµ-Optimality Conditions Limiting Operations The Bounded State Problem for Integrodifferential Systems The Bounded State Problem for Ordinary Differential Systems Further Discussion of the Bounded State Problem Sufficiency Conditions Nonlinear Beam Problem Hamilton-Jacobi Theory Introduction Problem Formulation and Assumptions Continuity of the Value Function The Lower Dini Derivate Necessary Condition The Value as Viscosity Solution Uniqueness The Value Function as Verification Function Optimal Synthesis The Maximum Principle Bibliography Index
Reviews for Nonlinear Optimal Control Theory
This book provides a thorough introduction to optimal control theory for nonlinear systems. ... The book is enhanced by the inclusion of many examples, which are analyzed in detail using Pontryagin's principle. ... An important feature of the book is its systematic use of a relaxed control formulation of optimal control problems. ... -From the Foreword by Wendell Fleming