# Quasilinearization and Invariant Imbedding

## 1st Edition

### With Applications to Chemical Engineering and Adaptive Control

**Authors:**E. Stanley Lee

**Editors:**Richard Bellman

**eBook ISBN:**9781483266756

**Imprint:**Academic Press

**Published Date:**1st January 1968

**Page Count:**350

## Description

Mathematics in Science and Engineering, Volume 41: Quasilinearization and Invariant Imbedding presents a study on the use of two concepts for obtaining numerical solutions of boundary-value problems—quasilinearization and invariant imbedding.

This book emphasizes that the invariant imbedding approach reformulates the original boundary-value problem into an initial value problem by introducing new variables or parameters, while the quasilinearization technique represents an iterative approach combined with linear approximations. This volume focuses on analytical aspects that are concerned with actual convergence rates and computational requirements, considering various efficient algorithms that are suited for various types of boundary-value problems.

This publication is a good reference for chemical and control engineers and scientists interested in obtaining numerical solutions of boundary-value problems in their particular fields.

## Table of Contents

Preface

Chapter 1. Introductory Concepts

1. Introduction

2. Quasilinearization

3. Invariant Imbedding

4. Invariant Imbedding versus the Classical Approach

5. Numerical Solution of Ordinary Differential Equations

6. Numerical Solution Terminologies

References

Chapter 2. Quasilinearization

1. Introduction

2. Nonlinear Boundary-Value Problems

3. Linear Boundary-Value Problems

4. Finite-Difference Method for Linear Differential Equations

5. Discussion

6. Newton-Raphson Method

7. Discussion

8. Quasilinearization

9. Discussion

10. Existence and Convergence

11. Existence

12. Convergence

13. Maximum Operation and Differential Inequalities

14. Construction of a Monotone Sequence

15. Approximation in Policy Space and Dynamic Programming

16. Discussion

17. Systems of Differential Equations

References

Chapter 3. Ordinary Differential Equations

1. Introduction

2. A Second-Order Nonlinear Differential Equation

3. Recurrence Relation

4. Computational Procedure

5. Numerical Results

6. Stability Problem in Numerical Solution—The Fixed Bed Reactor

7. Finite-Difference Method

8. Systems of Algebraic Equations Involving Tridiagonal Matrices

9. Numerical Results

10. Stability Problem with High Peclet Number

11. Adiabatic Tubular Reactor with Axial Mixing

12. Numerical Results

13. Discussion

14. Unstable Initial-Value Problems

15. Discussion

16. Systems of Differential Equations

17. Computational Considerations

18. Simultaneous Solution of Different Iterations

References

Chapter 4. Parameter Estimation

1. Introduction

2. Parameter Estimation and the "Black Box" Problem

3. Parameter Estimation and the Experimental Determination of Physical Constants

4. A Multipoint Boundary-Value Problems

5. The Least Squares Approach

6. Computational Procedure for a Simpler Problems

7. Numerical Results

8. Nonlinear Boundary Condition

9. Random Search Technique

10. Numerical Results

11. Discussion

12. Parameter Up-Dating

13. Discussion

14. Estimation of Chemical Reaction Rate Constants

15. Differential Equations with Variable Coefficients

16. An Example

17. III-Conditioned Systems

18. Numerical Results

19. Discussion

20. An Empirical Approximation

21. Numerical Results

22. A Second Approximation

23. Numerical Results

24. Differential Approximation

25. A Second Formulation

26. Computational Aspects

27. Discussion

References

Chapter 5. Optimization

1. Introduction

2. Optimum Temperature Profiles in Tubular Reactors

3. Numerical Results

4. Discussion

5. Back and Forth Integration

6. Two Consecutive Gaseous Reactions

7. Optimum Pressure Profile in Tubular Reactor

8. Numerical Results

9. Optimum Temperature Profile with Pressure as Parameter

10. Numerical Results and Procedures

11. Calculus of Variations with Control Variable Inequality Constraint

12. Calculus of Variations with Pressure Drop in the Reactor

13. Pontryagin's Maximum Principle

14. Discussion

15. Optimum Feed Conditions

16. Partial Derivative Evaluation

17. Conclusions

References

Chapter 6. Invariant Imbedding

1. Introduction

2. The Invariant Imbedding Approach

3. An Example

4. The Missing Final Condition

5. Determination of x and y in Terms of r and s

6. Discussion

7. Alternate Formulations—I

8. Linear and Nonlinear Systems

9. The Riccati Equation

10. Alternate Formulations—II

11. The Reflection and Transmission Functions

12. Systems of Differential Equations

13. Large Linear Systems

14. Computational Considerations

15. Dynamic Programming

16. Discussion

References

Chapter 7. Quasilinearization and Invariant Imbedding

1. Introduction

2. The Predictor-Corrector Formula

3. Discussion

4. Linear Boundary-Value Problems

5. Numerical Results

6. Optimum Temperature Profiles in Tubular Reactors

7. Numerical Results

8. Discussion

9. Dynamic Programming and Quasilinearization—I

10. Discussion

11. Linear Differential Equations

12. Dynamic Programming and Quasilinearization—II

13. Further Reduction in Dimensionality

14. Discussion

References

Chapter 8. Invariant Imbedding, Nonlinear Filtering, and the Estimation of Variables and Parameters

1. Introduction

2. An Estimation Problem

3. Sequential and Nonsequential Estimates

4. The Invariant Imbedding Approach

5. The Optimal Estimates

6. Equation for the Weighting Function

7. A Numerical Example

8. Systems of Differential Equations

9. Estimation of State and Parameter—An Example

10. A More General Criterion

11. An Estimation Problem with Observational Noise and Disturbance Input

12. The Optimal Estimate—A Two-Point Boundary-Value Problem

13. Invariant Imbedding

14. A Numerical Example

15. Systems of Equations with Observational Noises and Disturbance Inputs

16. Discussion

References

Chapter 9. Parabolic Partial Differential Equations—Fixed Bed Reactors with Axial Mixing

1. Introduction

2. Isothermal Reactor with Axial Mixing

3. An Implicit Difference Approximation

4. Computational Procedure

5. Numerical Results—Isothermal Reactor

6. Adiabatic Reactor with Axial Mixing

7. Numerical Results—Adiabatic Reactor

8. Discussion

9. Influence of the Packing Particles

10. The Linearized Equations

11. The Difference Equations

12. Computational Procedure—Fixed Bed Reactor

13. Numerical Results—Fixed Bed Reactor

14. Conclusion

References

Appendix I. Variational Problems with Parameters

1. Introduction

2. Variational Equations with Parameters

3. Simpler End Conditions

4. Calculus of Variations with Control Variable Inequality Constraint

5. Pontryagin's Maximum Principle

References

Appendix II. The Functional Gradient Technique

1. Introduction

2. The Recurrence Relations

3. Numerical Example

4. Discussion

References

Author Index

Subject Index

## Details

- No. of pages:
- 350

- Language:
- English

- Copyright:
- © Academic Press 1968

- Published:
- 1st January 1968

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483266756

## About the Author

### E. Stanley Lee

## About the Editor

### Richard Bellman

### Affiliations and Expertise

Departments of Mathematics, Electrical Engineering, and Medicine University of Southern California Los Angeles, California