Optimal Control for Chemical Engineers gives a detailed treatment of optimal control theory that enables readers to formulate and solve optimal control problems. With a strong emphasis on problem solving, the book provides all the necessary mathematical analyses and derivations of important results, including multiplier theorems and Pontryagin’s principle.
The text begins by introducing various examples of optimal control, such as batch distillation and chemotherapy, and the basic concepts of optimal control, including functionals and differentials. It then analyzes the notion of optimality, describes the ubiquitous Lagrange multipliers, and presents the celebrated Pontryagin principle of optimal control. Building on this foundation, the author examines different types of optimal control problems as well as the required conditions for optimality. He also describes important numerical methods and computational algorithms for solving a wide range of optimal control problems, including periodic processes.
Through its lucid development of optimal control theory and computational algorithms, this self-contained book shows readers how to solve a variety of optimal control problems.
Introduction
Definition
Optimal Control versus Optimization
Examples of Optimal Control Problems
Structure of Optimal Control Problems
Fundamental Concepts
From Function to Functional
Domain of a Functional
Properties of Functionals
Differential of a Functional
Variation of an Integral Objective Functional
Second Variation
Optimality in Optimal Control Problems
Necessary Condition for Optimality
Application to Simplest Optimal Control Problem
Solving an Optimal Control Problem
Sufficient Conditions
Piecewise Continuous Controls
Lagrange Multipliers
Motivation
Role of Lagrange Multipliers
Lagrange Multiplier Theorem
Lagrange Multiplier and Objective Functional
John Multiplier Theorem for Inequality Constraints
Pontryagin’s Minimum Principle
Application
Problem Statement
Pontryagin’s Minimum Principle
Derivation of Pontryagin’s Minimum Principle
Different Types of Optimal Control Problems
Free Final Time
Fixed Final Time
Algebraic Constraints
Integral Constraints
Interior Point Constraints
Discontinuous Controls
Multiple Integral Problems
Numerical Solution of Optimal Control Problems
Gradient Method
Penalty Function Method
Shooting Newton-Raphson Method
Optimal Periodic Control
Optimality of Periodic Controls
Solution Methods
Pi Criterion
Pi Criterion with Control Constraints
Mathematical Review
Limit of a Function
Continuity of a Function
Intervals and Neighborhoods
Bounds
Order of Magnitude
Tayor Series and Remainder
Autonomous Differential Equations
Differential
Derivative
Newton-Raphson Method
Fundamental Theorem of Calculus
Mean Value Theorem
Intermediate Value Theorem
Implicit Function Theorem
Bolzano-Weierstrass Theorem
Weierstrass Theorem
Linear or Vector Space
Direction of a Vector
Parallelogram Identity
Triangle Inequality for Integrals
CauchySchwarz Inequality
Operator Inequality
Conditional Statement
Fundamental Matrix
Index
Bibliography and Exercises appear at the end of each chapter.
Biography
Simant Ranjan Upreti is a professor of chemical engineering at Ryerson University in Toronto. His research interests include the mathematical modeling, computer simulation, optimization, and optimal control of chemical engineering processes. Dr. Upreti has been involved in the application of optimal control to determine concentration-dependent diffusion of gases in heavy oils and polymers and to enhance the recovery of heavy oils.
