Optimal Control Applied to Biological Models

by Lenhart; Suzanne

• ISBN13: 9781584886402
• ISBN10: 1584886404
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-05-07
• Publisher: Chapman & Hall/

Summary

From economics and business to the biological sciences to physics and engineering, professionals successfully use the powerful mathematical tool of optimal control to make management and strategy decisions. Optimal Control Applied to Biological Models thoroughly develops the mathematical aspects of optimal control theory and provides insight into the application of this theory to biological models.Focusing on mathematical concepts, the book first examines the most basic problem for continuous time ordinary differential equations (ODEs) before discussing more complicated problems, such as variations of the initial conditions, imposed bounds on the control, multiple states and controls, linear dependence on the control, and free terminal time. In addition, the authors introduce the optimal control of discrete systems and of partial differential equations (PDEs).Featuring a user-friendly interface, the book contains fourteen interactive sections of various applications, including immunology and epidemic disease models, management decisions in harvesting, and resource allocation models. It also develops the underlying numerical methods of the applications and includes the MATLAB® codes on which the applications are based.Requiring only basic knowledge of multivariable calculus, simple ODEs, and mathematical models, this text shows how to adjust controls in biological systems in order to achieve proper outcomes.

Table of Contents

Preface	p. xi
Basic Optimal Control Problems	p. 1
Preliminaries	p. 4
The Basic Problem and Necessary Conditions	p. 7
Pontryagin's Maximum Principle	p. 12
Exercises	p. 18
Existence and Other Solution Properties	p. 21
Existence and Uniqueness Results	p. 23
Interpretation of the Adjoint	p. 26
Principle of Optimality	p. 28
The Hamiltonian and Autonomous Problems	p. 31
Exercises	p. 35
State Conditions at the Final Time	p. 37
Payoff Terms	p. 37
States with Fixed Endpoints	p. 41
Exercises	p. 46
Forward-Backward Sweep Method	p. 49
Lab 1: Introductory Example	p. 57
Lab 2: Mold and Fungicide	p. 63
Lab 3: Bacteria	p. 67
Bounded Controls	p. 71
Necessary Conditions	p. 73
Numerical Solutions	p. 81
Exercises	p. 83
Lab 4: Bounded Case	p. 85
Lab 5: Cancer	p. 89
Lab 6: Fish Harvesting	p. 93
Optimal Control of Several Variables	p. 97
Necessary Conditions	p. 97
Linear Quadratic Regulator Problems	p. 104
Higher Order Differential Equations	p. 107
Isoperimetric Constraints	p. 108
Numerical Solutions	p. 112
Exercises	p. 113
Lab 7: Epidemic Model	p. 117
Lab 8: HIV Treatment	p. 123
Lab 9: Bear Populations	p. 129
Lab 10: Glucose Model	p. 135
Linear Dependence on the Control	p. 139
Bang-Bang Controls	p. 139
Singular Controls	p. 143
Exercises	p. 151
Lab 11: Timber Harvesting	p. 153
Lab 12: Bioreactor	p. 157
Free Terminal Time Problems	p. 163
Necessary Conditions	p. 163
Time Optimal Control	p. 168
Exercises	p. 173
Adapted Forward-Backward Sweep	p. 175
Secant Method	p. 175
One State with Fixed Endpoints	p. 177
Nonlinear Payoff Terms	p. 182
Free Terminal Time	p. 183
Multiple Shots	p. 184
Exercises	p. 187
Lab 13: Predator-Prey Model	p. 189
Discrete Time Models	p. 193
Necessary Conditions	p. 193
Systems Case	p. 199
Exercises	p. 202
Lab 14: Invasive Plant Species	p. 205
Partial Differential Equation Models	p. 211
Existence of an Optimal Control	p. 212
Sensitivities and Necessary Conditions	p. 213
Uniqueness of the Optimal Control	p. 215
Numerical Solutions	p. 215
Harvesting Example	p. 216
Beaver Example	p. 220
Predator-Prey Example	p. 223
Identification Example	p. 228
Controlling Boundary Terms	p. 231
Exercises	p. 234
Other Approaches and Extensions	p. 237
References	p. 245
Index	p. 259
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