Optimal Control

Preface
Preface to the Second Edition
Contents
Notations
Part I: Introduction
	Chapter 1: The Subject of Optimal Control
		1.1 «Mass-Spring» Example
		1.2 Subject and Problems of Optimal Control
		1.3 Place of Optimal Control
	Chapter 2: Mathematical Model for Controlled Object
		2.1 Controlled Object
		2.2 Control and Trajectory
		2.3 Mathematical Model
		2.4 Existence and Uniqueness of a Process
		2.5 Linear Models
		2.6 Example
Part II: Control of Linear Systems
	Chapter 3: Reachability Set
		3.1 Cauchy Formula
		3.2 Properties of the Fundamental Matrix
		3.3 Examples
		3.4 Definition of a Reachability Set
		3.5 Limitation and Convexity
			3.5.1 Limitation
			3.5.2 Convexity
		3.6 Closure
		3.7 Continuity
		3.8 Extreme Principle
		3.9 Application of the Extreme Principle
	Chapter 4: Controllability of Linear Systems
		4.1 Point-to-Point Controllability
		4.2 Analysis of the Point-to-Point Controllability Criteria
		4.3 Auxiliary Lemma
		4.4 Kalman Theorem
		4.5 Control with Minimal Norm
		4.6 Construction of Control with Minimum Norm
		4.7 Total Controllability of Linear System
		4.8 Synthesis of Control with a Minimal Norm
		4.9 Krasovskii Theorem
		4.10 Total Controllability of Stationary System
		4.11 Geometry of a Non-controllable System
		4.12 Transformation of Non-controllable System
		4.13 Controllability of Transformed System
	Chapter 5: Minimum Time Problem
		5.1 Statement of the Problem
		5.2 Existence of a Solution of the Minimum Time Problem
		5.3 Criterion of Optimality
		5.4 Maximum Principle for the Minimum Time Problem
		5.5 Stationary Minimum Time Problem
	Chapter 6: Synthesis of the Optimal System Performance
		6.1 General Scheme to Apply the Maximum Principle
		6.2 Control of Acceleration of a Material Point
		6.3 Concept of Optimal Control Synthesis
		6.4 Examples of Synthesis of Optimal Systems Performance
			6.4.1 Eigenvalues of Matrix A Are Real and Distinct
			6.4.2 The Eigenvalues of Matrix A Are Complex
	Chapter 7: The Observability Problem
		7.1 Statement of the Problem
		7.2 Criterion of Observability
		7.3 Observability in Homogeneous System
		7.4 Observability in Nonhomogeneous System
		7.5 Observability of an Initial State
		7.6 Relation Between Controllability and Observability
		7.7 Total Observability of a Stationary System
	Chapter 8: Identification Problem
		8.1 Statement of the Problem
		8.2 Criterion of Identifiability
		8.3 Restoring the Parameter Vector
		8.4 Total Identificaition of Stationary System
Part III: Control of Nonlinear Systems
	Chapter 9: Types of Optimal Control Problems
		9.1 General Characteristics
		9.2 Objective Functionals
			Terminal Functional (Mayer Functional)
			Mayer-Bolts Functional
		9.3 Constraints on the Ends of a Trajectory. Terminology
		9.4 The Simplest Problem
		9.5 Two-Point Minimum Time Problem
		9.6 General Optimal Control Problem
		9.7 Problem with Intermediate States
		9.8 Common Problem of Optimal Control
	Chapter 10: Small Increments of a Trajectory
		10.1 Statement of a Problem
		10.2 Evaluation of the Increment of Trajectory
		10.3 Representation of Small Increments of Trajectory
		10.4 Relation of the Ends of Trajectories
	Chapter 11: The Simplest Problem of Optimal Control
		11.1 Simplest-Problem. Functional Increment Formula
		11.2 Maximum Principle for the Simplest Problem
		11.3 Boundary Value Problem of the Maximum Principle
		11.4 Continuity of the Hamiltonian
		11.5 Sufficiency of the Maximum Principle
		11.6 Applying the Maximum Principle to the Linear Problem
		11.7 Solution of the Mass-Spring Example
	Chapter 12: General Optimal Control Problem
		12.1 General Problem. Functional Increment Formula
		12.2 Variation of the Process
		12.3 Necessary Conditions for Optimality
		12.4 Lagrange Multiplier Rule
		12.5 Universal Lagrange Multipliers
		12.6 Maximum Principle for the General Problem
		12.7 Comments
		12.8 Sufficiency of the Maximum Principle
		12.9 Maximum Principle for Minimum Time Problem
		12.10 Maximum Principle and Euler-Lagrange Equation
		12.11 Maximum Principle and Optimality of a Process
	Chapter 13: Problem with Intermediate States
		13.1 Problem with Intermediate State. Functional Increment Formula
		13.2 Preliminary Necessary Conditions of Optimality
		13.3 Maximum Principle for the Problem with an Intermediate State
		13.4 Discontinuous Systems
	Chapter 14: Extremals Field Theory
		14.1 Specifying of the Problem
		14.2 Field of Extremals
		14.3 Exact Formula for Large Increments of a Functional
		14.4 Sufficiency of the Maximum Principle
		14.5 Invariance of the Systems
		14.6 Examples of an Invariant System
	Chapter 15: Sufficient Optimality Conditions
		15.1 Common Problem of Optimal Control
		15.2 Basic Theorems
		15.3 Analytical Construction of the Controller
		15.4 Relation with Dynamic Programming
Conclusion
Appendix
	A.1. Elements of Multidimensional Geometry
		A.1.1. Finite-Dimensional Vector Space
		A.1.2. Geometric Objects in Rn
	A.2. Elements of Convex Analysis
		A.2.1. Separability of Sets
		A.2.2. Reference Plane
		A.2.3. Representation of a Convex Set
		A.2.4. Convex Hull of a Set
	A.3. Maximum and Minimum of a Function
		A.3.1. Properties of a Maximum and Minimum
		A.3.2. Continuity of a Maximum and Minimum
	A.4. Tasks and Solutions
		A.4.1. Tasks
			Fundamental Matrix
			Reachability Set
			Reference Plane
			Point-to-Point Controllability
			Total Controllability
			Minimum Time Problem
			Observation Problem
			Identification Problem
			Synthesis of Control
			Variants of Tasks. Tests
			Variants of Tasks. Verification of a Process on Optimality
		A.4.2. Examples of a Solution
			Point-to-Point Controllability
			Non-stationary System. Rank Criterion
			Non-stationary System. Krasovsky´s Theorem
			Stationary System. Total Controllability
			Minimum Time Problem
			S-Problem
			G-Problem
			Variational Problem
Bibliography