Optimal Control Applied to Biological Models: 1st Edition (Hardback) book cover

Optimal Control Applied to Biological Models

1st Edition

By Suzanne Lenhart, John T. Workman

Chapman and Hall/CRC

274 pages | 42 B/W Illus.

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Hardback: 9781584886402
pub: 2007-05-07
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Description

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.

Reviews

". . . 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 Mirica, in Mathematical Reviews, 2008f

Table of Contents

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

Interpretation of the Adjoint

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

Linear Quadratic Regulator Problems

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

ADAPTED FORWARD-BACKWARD SWEEP

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

About the Series

Chapman & Hall/CRC Mathematical and Computational Biology

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI008000
SCIENCE / Life Sciences / Biology / General