An Introduction to Optimal Control Applied to Disease Models - - PowerPoint PPT Presentation

An Introduction to Optimal Control Applied to Disease Models

Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics

Lecture1 – p.1/37

Example

• ✁✄✂

Number of cancer cells at time

(exponential growth) State

Drug concentration Control

• ✝

• ✁✄✂

• ✁☛✡

known initial data minimize

• ✁

where the first term represents number of cancer cells and the second term represents harmful effects of drug on body.

Lecture1 – p.2/37

Optimal Control

Adjust controls in a system to achieve a goal System: Ordinary differential equations Partial differential equations Discrete equations Stochastic differential equations Integro-difference equations

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Deterministic Optimal Control

Control of Ordinary Differential Equations (DE)

control

state State function satisfies DE Control affects DE

Goal (objective functional)

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Basic Idea

System of ODEs modeling situation Decide on format and bounds on the controls Design an appropriate objective functional Derive necessary conditions for the optimal control Compute the optimal control numerically

Lecture1 – p.5/37

Design an appropriate objective functional –balancing opposing factors in functional –include (or not) terms at the final time

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Big Idea

In optimal control theory, after formulating a problem appropriate to the scenario, there are several basic problems : (a) to prove the existence of an optimal control, (b) to characterize the optimal control, (c) to prove the uniqueness of the control, (d) to compute the optimal control numerically, (e) to investigate how the optimal control depends on various parameters in the model.

Lecture1 – p.7/37

Deterministic Optimal Control- ODEs

Find piecewise continuous control

and associated state variable

to maximize

subject to

and

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Contd.

Optimal Control

achieves the maximum Put

into state DE and obtain

corresponding optimal state

,

• ptimal pair

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Necessary, Sufficient Conditions

Necessary Conditions If

,

are optimal, then the following conditions hold . . . Sufficient Conditions If

,

and

(adjoint) satisfy the conditions . . . then

,

are optimal.

Lecture1 – p.10/37

Adjoint

like Lagrange multipliers to attach DE to objective func- tional.

Lecture1 – p.11/37

Deterministic Optimal Control- ODEs

Find piecewise continuous control

and associated state variable

to maximize

subject to

and

Lecture1 – p.12/37

Quick Derivation of Necessary Condition

Suppose

is an optimal control and

corresponding state.

variation function,

.

t u*(t)+ ah(t) u*(t)

another control.

state corresponding to

,

Lecture1 – p.13/37

Contd.

At

,

t x * (t) y(t,a) x 0

all trajectories start at same position

when

control

Maximum of

w.r.t.

• ccurs at

.

Lecture1 – p.14/37

Contd.

Adding

to our

gives

Lecture1 – p.15/37

Contd.

here we used product rule and

.

Lecture1 – p.16/37

Contd.

Arguments of

terms are

.

Arguments of

terms are

.

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Contd.

Choose

s.t.

adjoint equation

transversality condition

arbitrary variation

for all

Optimality condition.

Lecture1 – p.18/37

Using Hamiltonian

Generate these Necessary conditions from Hamiltonian

integrand (adjoint) (RHS of DE) maximize w.r.t.

at

• ptimality eq.

adjoint eq.

transversality condition

Lecture1 – p.19/37

Converted problem of finding control to maximize

• bjective functional subject to DE, IC to using

Hamiltonian pointwise. For maximization

at

as a function of

For minimization

at

as a function of

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Two unknowns

and

introduce adjoint

• (like a Lagrange multiplier)

Three unknowns

,

and

• nonlinear w.r.t.

Eliminate

by setting

and solve for

in terms of

and

• Two unknowns

and

• with 2 ODEs (2 point BVP)

+ 2 boundary conditions.

Lecture1 – p.21/37

Pontryagin Maximum Principle

If

and

are optimal for above problem, then there exists adjoint variable

s.t.

at each time, where Hamiltonian

is defined by

and

transversality condition

Lecture1 – p.22/37

Hamiltonian

maximizes

w.r.t.

,

is linear w.r.t.

bounded controls,

. Bang-bang control or singular control Example:

cannot solve for

is nonlinear w.r.t.

, set

and solve for

• ptimality equation.

Lecture1 – p.23/37

Example 1

subject to

• ❊

• ❑

What optimal control is expected?

Lecture1 – p.24/37

Example 1 worked

integrand

RHS of DE

at

Lecture1 – p.25/37

Example 2

subject to

What optimal control is expected?

Lecture1 – p.26/37

Hamiltonian

The adjoint equation and transversality condition give

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Example 2 continued

and the optimality condition leads to

The associated state is

Lecture1 – p.28/37

Graphs, Example 2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 Time State 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 4 6 8 Time Adjoint 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −8 −6 −4 −2 Time Control Example 2.2

Lecture1 – p.29/37

Example 3

at

Lecture1 – p.30/37

Contd.

Solve for

and then get

. Do numerically with Matlab or by hand

Lecture1 – p.31/37

Exercise

control

Lecture1 – p.32/37

Exercise completed

• ✁

• ☎

control

• ☎

• ☞

• ☎

• ✞

• ☎

• ý

• ☎

• ✕

Lecture1 – p.33/37

Contd.

There is not an “Optimal Control" in this case. Want finite maximum. Here unbounded optimal state unbounded OC

Lecture1 – p.34/37

Opening Example

Number of cancer cells at time

(exponential growth) State

Drug concentration Control

known initial data minimize

See need for bounds on the control. See salvage term.

Lecture1 – p.35/37

Further topics to be covered

Interpretation of the adjoint Salvage term Numerical algorithms Systems case Linear in the control case Discrete models

Lecture1 – p.36/37

more info

See my homepage www.math.utk.edu

lenhart Optimal Control Theory in Application to Biology short course lectures and lab notes Book: Optimal Control applied to Biological Models CRC Press, 2007, Lenhart and J. Workman

Lecture1 – p.37/37