1 optimal control for integrodifferencequa tions andrew whittle university of tennessee department...
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Optimal control for
integrodifferencequations
Optimal control for
integrodifferencequations
Andrew WhittleUniversity of Tennessee
Department of Mathematics
Andrew WhittleUniversity of Tennessee
Department of Mathematics
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Outline
• Including space
➡ Cellular Automata, Coupled Lattice maps, Integrodifference equations
• Integrodifference models
➡ Description, dispersal kernels
• Optimal control of Integrodifference models
➡ Set up of the bioeconomic model
➡ Application for gypsy moths
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spacespace
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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
Optimal control for integrodifference
equations
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Example
• Gypsy moths are a forest pest with cyclic population levels
• Larvae eat the leaves of trees causing extensive defoliation across northeastern US (13 million acres in 1981)
• This leaves the trees weak and vulnerable to disease
• Potential loss of a Oak species
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life cycle of the gypsy moth
Egg mass
Adult
Larvae
Pupa
NPVInfectedLarvae
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NPV
• Nuclear Polyhedrosis Virus (NPV) is a naturally occurring virus
• 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 fecundityReduction in gypsy moth susceptibility to
NPV
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model for gypsy moths
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Biocontrol
• Natural enemy or enemies, typically from the intruder’s native region, is introduced to keep the pest under control
• More biocontrol agent (NPV) can be produced (Gypchek)
• Production is labor intensive and therefore costly but it is still used by USDA to fight major outbreaks
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bioeconomic modelWe 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
• Suzanne, Joshi and Holly developed optimal control theory for integrodifference equations
• 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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Finding the adjoint functions allows us to replace the sensitivities in the directional derivative of the objective function
Adjoints
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Including the control bounds we get the optimal control
By a change of order of integration we have
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numerical method
• Starting guess for control values
State equationsforward
Adjoint equationsbackward
Updatecontrols
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results
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Total Cost, J(p)
B,
J(p)
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Summary
• Integrodifference equations are a useful tool in modeling populations with discrete non-overlapping generations
• Optimal control of Integrodifference equations is a new growing area and has practical applications
• For gypsy moths, optimal solutions suggest a longer period of application in low gypsy moth density areas
• Critical range of Bn (balancing coefficient) that cause a considerable decrease in Total costs
• More work needs to be done for example, in cost functions for the objective function