**Optimal Control for Chemical Engineers**Â 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 Pontryaginâ€™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.

#### Table of Contents – Optimal Control for Chemical Engineers

**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

**Pontryaginâ€™s Minimum Principle**Application

Problem Statement

Pontryaginâ€™s Minimum Principle

Derivation of Pontryaginâ€™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.*

### About the Author

**Simant Ranjan Upreti**Â 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.