"Two Biological Applications of Optimal Control to Hybrid Differential Equations and Elliptic Partial Differential Equations"
Dr. Suzanne Lenhart
- Population and Disease Models (with seasonality)
Optimal Control on Hybrid ODE Systems with Application to a Tick Disease Model
------ accepted by Mathematical Biosciences and Engineering, June 2007.
We are considering an optimal control problem for a type of hybrid system involving ordinary
differential equations and some discrete time feature. One state variable has dynamics in only one season of the year and has a jump condition to
obtain the initial condition for that corresponding season in the next year. The other state variable has continuous dynamics. Given a general
objective functional, existence, necessary conditions and uniqueness for an optimal control are established. We apply our approach to a
tick-transmitted disease model in which ticks dynamics changes in a seasonal way while hosts have continuous dynamics. The goal is to maximize
disease-free ticks and minimize infected ticks through
an optimal control strategy of treatment with acaricide. Numerical examples are given to illustrate the results.
- Natural Resource Modeling
Optimal Harvesting of a Spatially Explicit Fishery Model
Wandi Ding and Suzanne Lenhart
------ submitted to Natural Resource Modeing, June 2007.
We consider an optimal fishery harvesting problem using a
semilinear elliptic PDE model with Dirichlet boundary conditions,
which has logistic population growth and spatially explicit
harvesting. We consider two objective functionals: maximizing the
yield and minimizing the cost or the variation in the fishing
effort (control). Existence, necessary conditions and uniqueness
for the optimal harvesting control for both cases are established.
Results for maximizing the yield with no-flux boundary conditions
are also given. The optimal control when minimizing the variation
is characterized by a variational inequality instead of the usual
algebraic characterization, which involves the solutions of an
optimality system of nonlinear elliptic partial differential
equations. Numerical examples are given to illustrate the
- Computational Science Workshop for Natural Resource Managers 2007
- Computational Science Workshop for Natural Resource Managers 2006
- Optimal Control on Discrete Model
Rabies in Raccoons: Optimal Control for a Discrete Time Model
on a Spatial Grid
Wandi Ding, Louis Gross, Keith Langston, Suzanne Lenhart and Leslie A. Real
------ accepted by Journal of Biological Dynamics, June 2007.
An epidemic model for rabies in raccoons is formulated with
discrete time and spatial features. The goal is to analyze the
strategies for optimal distribution of vaccine baits to minimize
the spread of the disease and the cost of implementing the
control. Discrete optimal control techniques are used to derive
the optimality system, which is then solved numerically to
illustrate various scenarios.
- Discrete Model for Cod with Age Structure (with H. Behncke: Universitat Osnabrueck, Germany)
- Population Model with Control on the Growth Coefficient
(with S. Lenhart, H. Finotti: University of Tennessee, Y. Ye: Shanghai University of Finance & Economics, China and Y. Lou: Ohio State University)
- Pest Control (Bio-control)
- Industrial Mathematical & Statistical Modeling Workshop @ NC State Univ. 2004
Natalie Almond, Wandi Ding, Xiaochuan Li, Xingtao Liu, Steven Rusnica, Ismael
Velzquez-Ramrez, Emily Lada, Fazafumi Ito, Michael Horton, Mellisa Choi
"Mobile Sensing of Aerosolized Chemical and Biological Agents"
------ CRSC Technical Report, CRSC-TR04-41 (p.15-26), Dec. 2004
- Shusen Xie and Wandi Ding
"Linear Modified Finite Difference Method Combined with Characteristics for Convection-Diffusion Problems"
------ J. Qingdao University of Science & Technology, No.2, 9-13, 2002
Linear modified characteristic finite difference and alternating direction schemes are devised for nonlinear convection diffusion
problems. The advantage of these schemes is that they reduce nonlinear problems algebraically to solving at each time step the same
linear system with different right hand side. The alternating direction schemes can reduce the large multi dimensional problems
to a series of smaller one dimensional ones. Stability and optimal order of discrete L2 error estimates are derived.
- M.S. Thesis
"Linear Modified Finite Difference Method for a Type of Initial Value Problems of Parabolic Partial Differential Equations"
------ Ocean University of Qingdao, China, 2001, Library collection code: Y406018
- Shusen Xie and Wandi Ding
"Optimal Numerical Primitive Function Formula in the space W(p;B)" (Preprint, 2002)