Optimal Control Applied to Biological Models

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ISBN 9781584886402
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Features

  • Provides the theoretical foundation to successfully control various types of biological systems
  • Covers a range of biological applications, including bacteria, cancer, HIV treatment, glucose, and bioreactor models
  • Includes MATLAB codes throughout the text and on the Web, enabling readers to modify the codes for their own applications
  • Presents numerous examples and exercises that illustrate how to solve optimal control problems involving systems of ODEs and bang bang and singular controls
  • 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

    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

    Editorial 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