Introductory Lecture 2 Suzanne Lenhart University of Tennessee, - - PowerPoint PPT Presentation

Introductory Lecture 2

Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics Supported by NIH grant

Lecture2 – p.1/25

Salvage Term

• ✁

where

is final payoff. What change results?

. . .

same as before

Lecture2 – p.2/25

Only change

Example

Lecture2 – p.3/25

Example

Number of cancer cells at time

(exponential growth)

State

Drug concentration Control

known initial data

• ■❍

where the first term is number of cancer cells at final time

and the second term is the harmful effects of drug on body.

Lecture2 – p.4/25

at

transversality condition

here

Lecture2 – p.5/25

Contd.

Lecture2 – p.6/25

Well Stirred Bioreactor

Contaminant and bacteria present in spatially uniform time varying concentrations

concentration of contaminant

concentration of bacteria bioreactor rich in all nutrients except one

concentration of input nutrient

bacteria degrades contaminant via co-metabolism.

Lecture2 – p.7/25

where

where

is control and

are known. Objective functional:

Find

to maximize

maximize bacteria and minimize input nutrient cost.

Lecture2 – p.8/25

penalizes large values of

at final time . Can eliminate

variable and work with

• nly.

Lecture2 – p.9/25

at

Lecture2 – p.10/25

known

Solve for

numerically.

Lecture2 – p.11/25

Problems

What if:

Need additional constraint

Lecture2 – p.12/25

Fishery Model

• ù

population level of fish

• ý✂✁

harvesting control Maximizing net profit:

• ù

• ù

• ✎

where

is discount factor,

terms represent profit from sale of fish, diminishing returns when there is a large amount of fish to sell and cost of fishing.

are positive constants.

Lecture2 – p.13/25

Contd.

Lecture2 – p.14/25

Contd.

Solve for

numerically. Need control bounds

Ref: B D Craven book Control and Optimization

Lecture2 – p.15/25

Interpretation of Adjoint

• ❇

( Definition of value function )

Units: money/unit item in profit problems.

Lecture2 – p.16/25

= marginal variation in the optimal objective functional value of the state value at

. “Shadow price”

additional money associated with addi- tional increment of the state variable

for all

“If one fish is added to the stock, how much is the value of the fishery affected ?"

Lecture2 – p.17/25

Approximate

New value Original value + adjoint

Lecture2 – p.18/25

Principle of Optimality

If

is an optimal pair on

and

, then

is also optimal for the problem on

:

t x 0 t0 t t1 (t, x*(t )) x

Lecture2 – p.19/25

Existence of Optimal Controls

“Sufficient conditions to guarantee existence of OC" Suppose

satisfy

is maximized w.r.t.

at

plus set of controls compact

jointly concave in

and

bounded state functions

Lecture2 – p.20/25

For details about existence of OC see Macki and Strauss book Fleming and Rishel book

Back to exercise example

To guarantee the maximum value of

would be finite, need a priori bound on state

, control

.

Lecture2 – p.21/25

Optimality System

State system coupled with adjoint system

• optimal control’s expressions substituted in

Uniqueness of Optimality System

Uniqueness of Optimal Control Uniqueness of Optimality System - only for small time

due to opposite time orientations BUT Uniqueness of Optimal Control

Uniqueness of Solutions of Optimality System To get uniqueness of OC directly, need strict concavity of

.

Lecture2 – p.22/25

Optimality System

State system coupled with adjoint system

• optimal control’s expressions substituted in

Uniqueness of Optimality System - only for small time

due to opposite time orientations Numerical Solutions by Iterative Method

• with Runge Kutta 4, Matlab or favorite ODE solver

(Characterization of OC non-smooth)

• guess for controls, solve forward for states
• solve backward for adjoints
• update controls, using characterization
• repeat forward and backwards sweeps and control

updates until convergence of iterates

Lecture2 – p.23/25

Idea of Runge Kutta

Give handouts.

Lecture2 – p.24/25

2 BC on one state

For Lab example Suppose

and

are BOTH GIVEN Then

doesnot have a boundary condition. Needs a type of shooting method.

Lecture2 – p.25/25