Transcript Optimal Control for Integrodifference Equations

Optimal control for
integrodifferencequa
tions
Andrew Whittle
University of Tennessee
Department of Mathematics
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Outline
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Including space
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Integrodifference models
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Cellular Automata, Coupled Lattice maps,
Integrodifference equations
Description, dispersal kernels
Optimal control of Integrodifference models
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Set up of the bioeconomic model
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Application for gypsy moths
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space
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Cellular Automata
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Coupled lattice maps
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Integrodifference equations
Integrodifference
equations are discrete
in one variable (usually
time) and continuous
in another (usually
space)
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Dispersal data
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Optimal control for
integrodifference
equations
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Example
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Gypsy moths are a forest pest with cyclic
population levels
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Larvae eat the leaves of trees causing
extensive defoliation across northeastern
US (13 million acres in 1981)
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This leaves the trees weak and vulnerable
to disease
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Potential loss of a Oak species
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life cycle of the gypsy moth
Egg mass
Adult
NPV
Infected
Larvae
Larvae
Pupa
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NPV
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Nuclear Polyhedrosis Virus (NPV) is a
naturally occurring virus
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Virus is specific to Gypsy moths and
decays from ultraviolet light
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tannin
Tannin is a chemical produced be plants to defend
itself from severe defoliation
Reduction in gypsy moth fecundity
Reduction in gypsy moth susceptibility to NPV
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model for gypsy moths
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Biocontrol
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Natural enemy or enemies, typically from the
intruder’s native region, is introduced to keep
the pest under control
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More biocontrol agent (NPV) can be
produced (Gypchek)
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Production is labor intensive and therefore
costly but it is still used by USDA to fight
major outbreaks
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bioeconomic model
We form an objective function that we wish to
minimize
where the control belong to the bounded set
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optimal control of integrodifference
equations
We wish to minimize the objective function
subject to the constraints of the state system
We first prove the existence and uniqueness of the
optimal control
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There is no Pontryagin’s maximum
principle for integrodifference equations
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Suzanne, Joshi and Holly developed
optimal control theory for
integrodifference equations
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This uses ideas from the discrete
maximum principle and optimal control
of PDE’s
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Characterization of the
optimal control
We take directional derivatives of the objective
functional
In order to do this we must first differentiate the
state variables with respect to the control
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Sensitivities and adjoints
By differentiating the state system we get the sensitivity
equations
The sensitivity system is linear. From the sensitivity
system we can find the adjoints
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Adjoints
Finding the adjoint functions allows us to replace the
sensitivities in the directional derivative of the objective
function
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By a change of order of integration we have
Including the control bounds we get the optimal control
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numerical method
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Starting guess for control values
State equations
forward
Update
controls
Adjoint equations
backward
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results
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J(p)
B,
Total Cost, J(p)
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Summary
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Integrodifference equations are a useful tool in modeling
populations with discrete non-overlapping generations
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Optimal control of Integrodifference equations is a new growing
area and has practical applications
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For gypsy moths, optimal solutions suggest a longer period of
application in low gypsy moth density areas
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Critical range of Bn (balancing coefficient) that cause a
considerable decrease in Total costs
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More work needs to be done for example, in cost functions for
the objective function
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