We apply discrete time optimal control theory to the mathematical modeling of pest control. Two scenarios: biological control and the combination of pesticide and biological control are considered. The goal is maximizing the "valuable" population, minimizing the pest population and the cost to apply the control strategies. Using the extension of Pontryagin's maximum principle to discrete system, the adjoint systems and the characterization of the optimal pest controls are derived. Numerical simulations of various cases are provided to show the effectiveness of our methods.

As an extension of the works of Coggeshall 1 and Ramsey 12 , a class of analytic solutions to the radiation hydrodynamics equations is derived for code verification purposes. These solutions are valid under assumptions including diffusive radiation transport, a polytropic gas equation of state, constant conductivity, separable flow velocity proportional to the curvilinear radial coordinate, and divergence-free heat flux. In accordance with these assumptions, the derived solution class is mathematically invariant with respect to the presence of radiative heat conduction, and thus represents a solution to the compressible flow (Euler) equations with or without conduction terms included. With this solution class, a quantitative code verification study (using spatial convergence rates) is performed for the cell-centered, finite volume, Eulerian compressible flow code xRAGE developed at Los Alamos National Laboratory. Simulation results show near second order spatial convergence in all physical variables when using the hydrodynamics solver only, consistent with that solver's underlying order of accuracy. However, contrary to the mathematical properties of the solution class, when heat conduction algorithms are enabled the calculation does not converge to the analytic solution.

In 1991, Coggeshall published a series of 22 closed-form solutions of the Euler compressible flow equations with a heat conduction term included. A remarkable feature of some of these solutions is invariance with respect to conduction; this phenomenon follows from subtle ancillary constraints wherein a heat flux term is assumed to be either identically zero or nontrivially divergence-free. However, the solutions featuring the nontrivial divergence-free heat flux constraint can be shown to be incomplete, using a well-known result most commonly encountered in elementary electrostatic theory. With this result, the application of the divergence operator to the heat flux distributions exhibited by many of the solutions yields a delta function source term instead of identically zero. In theory, the relevant solutions will be conduction invariant only if the appropriate source term is included. This result has important implications for the use of the Coggeshall similarity solutions as code verification test problems for simulation codes featuring coupled compressible fluid flow and heat conduction processes. Computational reproduction of the conduction invariance property represents a conceptually simple check for verifying the robustness of a multiphysics algorithm. In this work, it is demonstrated in the context of various computational instantiations of Coggeshall solution #8 (Cog8) that to maintain any semblance of conduction invariance, a heat source term must be included even with a simple nonlinear heat conduction process. The efficacy of the heat source term is shown to depend not only on values of the various free parameters included in the Coggeshall mathematical model but also the representation of heat sources in multiphysics simulation codes of interest.

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