1st Edition

# Optimal Control Applied to Biological Models

By Suzanne Lenhart, John T. Workman Copyright 2007
274 Pages 42 B/W Illustrations
by Chapman & Hall

274 Pages
by Chapman & Hall

Also available as eBook on:

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.

BASIC OPTIMAL CONTROL PROBLEMS
Preliminaries
The Basic Problem and Necessary Conditions
Pontryagin's Maximum Principle
Exercises

EXISTENCE AND OTHER SOLUTION PROPERTIES
Existence and Uniqueness Results
Principle of Optimality
The Hamiltonian and Autonomous Problems
Exercises

STATE CONDITIONS AT THE FINAL TIME
Payoff Terms
States with Fixed Endpoints
Exercises

FORWARD-BACKWARD SWEEP METHOD

LAB 1: INTRODUCTORY EXAMPLE

LAB 2: MOLD AND FUNGICIDE

LAB 3: BACTERIA

BOUNDED CONTROLS
Necessary Conditions
Numerical Solutions
Exercises

LAB 4: BOUNDED CASE

LAB 5: CANCER

LAB 6: FISH HARVESTING

OPTIMAL CONTROL OF SEVERAL VARIABLES
Necessary Conditions
Higher Order Differential Equations
Isoperimetric Constraints
Numerical Solutions
Exercises

LAB 7: EPIDEMIC MODEL

LAB 8: HIV TREATMENT

LAB 9: BEAR POPULATIONS

LAB 10: GLUCOSE MODEL

LINEAR DEPENDENCE ON THE CONTROL
Bang-Bang Controls
Singular Controls
Exercises

LAB 11: TIMBER HARVESTING

LAB 12: BIOREACTOR

FREE TERMINAL TIME PROBLEMS
Necessary Conditions
Time Optimal Control
Exercises

Secant Method
One State with Fixed Endpoints
Nonlinear Payoff Terms
Free Terminal Time
Multiple Shots
Exercises

LAB 13: PREDATOR-PREY MODEL

DISCRETE TIME MODELS
Necessary Conditions
Systems Case
Exercises

LAB 14: INVASIVE PLANT SPECIES

PARTIAL DIFFERENTIAL EQUATION MODELS
Existence of an Optimal Control
Sensitivities and Necessary Conditions
Uniqueness of the Optimal Control
Numerical Solutions
Harvesting Example
Beaver Example
Predator-Prey Example
Identification Example
Controlling Boundary Terms
Exercises

OTHER APPROACHES AND EXTENSIONS

REFERENCES

INDEX

### Biography

Suzanne Lenhart, John T. Workman

". . . the present book has the merit of collecting and treating in a unitary and accessible manner a large number of relevant problems in mathematical biology; the text is well written, systematically presented, accurate most of the time and accessible to a fairly large audience; it could do a great service to the community of researchers in mathematical control theory . . ."

– Stefan Mirică, in Mathematical Reviews, 2008f