Optimal Control Applied to Biological Models

This text introduces the mathematical aspects of optimal control theory and provides insight into the application of this theory to biological models. It examines various problems, such as the continuous time ordinary differential equation, variations of the initial conditions, linear dependence on the control, and free terminal time. The book also features a range of biological applications, including immunology and epidemic disease models, management decisions in harvesting, and resource allocation models. With numerous examples and exercises, the text includes MATLAB codes throughout, enabling readers to modify the codes for their own applications.

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 ExercisesEXISTENCE AND OTHER SOLUTION PROPERTIES Existence and Uniqueness Results Interpretation of the Adjoint Principle of Optimality The Hamiltonian and Autonomous Problems ExercisesSTATE CONDITIONS AT THE FINAL TIME Payoff TermsStates with Fixed EndpointsExercisesFORWARD-BACKWARD SWEEP METHOD LAB 1: INTRODUCTORY EXAMPLE LAB 2: MOLD AND FUNGICIDELAB 3: BACTERIABOUNDED CONTROLS Necessary Conditions Numerical Solutions ExercisesLAB 4: BOUNDED CASE LAB 5: CANCER LAB 6: FISH HARVESTINGOPTIMAL CONTROL OF SEVERAL VARIABLES Necessary Conditions Linear Quadratic Regulator Problems Higher Order Differential EquationsIsoperimetric Constraints Numerical Solutions Exercises LAB 7: EPIDEMIC MODEL LAB 8: HIV TREATMENT LAB 9: BEAR POPULATIONS LAB 10: GLUCOSE MODELLINEAR DEPENDENCE ON THE CONTROL Bang-Bang ControlsSingular Controls Exercises LAB 11: TIMBER HARVESTINGLAB 12: BIOREACTOR FREE TERMINAL TIME PROBLEMS Necessary Conditions Time Optimal Control ExercisesADAPTED FORWARD-BACKWARD SWEEP Secant Method One State with Fixed EndpointsNonlinear 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 SPECIESPARTIAL DIFFERENTIAL EQUATION MODELS Existence of an Optimal ControlSensitivities 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 REFERENCESINDEX