Optimal Control of Discrete Models Suzanne Lenhart University of - - PowerPoint PPT Presentation

Optimal Control of Discrete Models

Suzanne Lenhart University of Tennessee, Knoxville Department of Mathematics

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Introduction

For many populations, births and growth occur in regular times each year (or each cycle). Discrete time models are well suited to describe the life histories of organisms with discrete reproduction and/or growth.

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Another example

For example, the Beverton-Holt stock-recruitment model for a population

• at time

is

• ✂

• ✝
• ✆

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Age structured models

A population may be divided into separate discrete age classes. At each time step, a certain proportion of each class may survive and enter the next age class. Individuals in the first age class originate through reproduction from other classes.

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Simple Control Problem

subject to

for

state

control

, one less component

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Control Problem

Given a control

and initial state

, the state equation is given by the difference equation

for

. Note that the state has

• ne more component than the control

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Goal

• ❍❏■

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Hamiltonian

for

Notice the indexing on the adjoint. Necessary conditions

at

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Simple Control Problem

subject to

for

The control is the input (growth or decay). What optimal control is expected? (min state and size of control)

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Optimality Conditions

Our necessary conditions are

for

at

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continuing

for

and

solve algebraic equations

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Control Answers

• ptimal state values

,

• ptimal control values

,

,

.

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System Case

for

.

There are

controls,

states, and time steps. Define the objective functional as

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continued

at

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Simple pest control problem

good population and

pest population to be controlled with

.

• ó

for

for

for

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