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.
". . . 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
BASIC OPTIMAL CONTROL PROBLEMS
The Basic Problem and Necessary Conditions
Pontryagin's Maximum Principle
EXISTENCE AND OTHER SOLUTION PROPERTIES
Existence and Uniqueness Results
Interpretation of the Adjoint
Principle of Optimality
The Hamiltonian and Autonomous Problems
STATE CONDITIONS AT THE FINAL TIME
States with Fixed Endpoints
FORWARD-BACKWARD SWEEP METHOD
LAB 1: INTRODUCTORY EXAMPLE
LAB 2: MOLD AND FUNGICIDE
LAB 3: BACTERIA
LAB 4: BOUNDED CASE
LAB 5: CANCER
LAB 6: FISH HARVESTING
OPTIMAL CONTROL OF SEVERAL VARIABLES
Linear Quadratic Regulator Problems
Higher Order Differential Equations
LAB 7: EPIDEMIC MODEL
LAB 8: HIV TREATMENT
LAB 9: BEAR POPULATIONS
LAB 10: GLUCOSE MODEL
LINEAR DEPENDENCE ON THE CONTROL
LAB 11: TIMBER HARVESTING
LAB 12: BIOREACTOR
FREE TERMINAL TIME PROBLEMS
Time Optimal Control
ADAPTED FORWARD-BACKWARD SWEEP
One State with Fixed Endpoints
Nonlinear Payoff Terms
Free Terminal Time
LAB 13: PREDATOR-PREY MODEL
DISCRETE TIME MODELS
LAB 14: INVASIVE PLANT SPECIES
PARTIAL DIFFERENTIAL EQUATION MODELS
Existence of an Optimal Control
Sensitivities and Necessary Conditions
Uniqueness of the Optimal Control
Controlling Boundary Terms
OTHER APPROACHES AND EXTENSIONS
This series aims to capture new developments in mathematical biology, as well as high quality work summarizing or contributing to more established topics. Publishing a broad range of textbooks, reference works, and handbooks, the series is designed to appeal to students, researchers, and professionals in mathematical biology, as well as interdisciplinary researchers involved in associated fields.