392 Pages 12 B/W Illustrations
    by Chapman & Hall

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    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
    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
    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
    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
    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
    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
    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
    Penalty Proof of Necessary Conditions in Finite Dimensions
    The Norm of a Relaxed Control; Compact Constraints
    Necessary Conditions for an Unconstrained Problem
    The ε-Problem
    The ε-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

    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
    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
    Problem Statement and Assumptions
    Minimum Principle
    Some Linear Systems
    Linear Plant-Quadratic Cost
    Infinite Dimensional Setting

    Bounded State Problems
    Statement of the Problem
    ε-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
    Problem Formulation and Assumptions
    Continuity of the Value Function
    The Lower Dini Derivate Necessary Condition
    The Value as Viscosity Solution
    The Value Function as Verification Function
    Optimal Synthesis
    The Maximum Principle




    Leonard David Berkovitz, Negash G. Medhin

    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

    … more than a very useful research account and a handy reference to users of the theory-they also make it a pleasant and helpful study opportunity to students and other newcomers to the theory of optimal control.
    —Zvi Artstein, in Mathematical Reviews