OPTIMAL RESOURCE ALLOCATION FOR A DIFFUSIVE POPULATION MODEL

The spatial distribution of resources for diffusive populations can have a strong effect on population abundance. We investigate the optimal allocation of resources for a diffusive population. Population dynamics are represented by a parabolic partial differential equation with density-dependent growth and resources are represented through their space- and time-varying influence on the growth function. We consider both local and integral constraints on resource allocation. The goal is to maximize the abundance of the population while minimizing the cost of resource allocation. After characterizing the optimal control in terms of the population solution and the adjoint functions, we illustrate several scenarios numerically. The effects of initial and boundary conditions are important for the optimal allocation of resources.

First-Order Comprehensive Adjoint Method for Computing Operator-Valued Response Sensitivities to Imprecisely Known Parameters, Internal Interfaces and Boundaries of Coupled Nonlinear Systems: II. Application to a Nuclear Reactor Heat Removal Benchmark

This work illustrates the application of a comprehensive first-order adjoint sensitivity analysis methodology (1st-CASAM) to a heat conduction and convection analytical benchmark problem which simulates heat removal from a nuclear reactor fuel rod. This analytical benchmark problem can be used to verify the accuracy of numerical solutions provided by software modeling heat transport and fluid flow systems. This illustrative heat transport benchmark shows that collocation methods require one adjoint computation for every collocation point while spectral expansion methods require one adjoint computation for each cardinal function appearing in the respective expansion when recursion relations cannot be developed between the corresponding adjoint functions. However, it is also shown that spectral methods are much more efficient when recursion relations provided by orthogonal polynomials make it possible to develop recursion relations for computing the corresponding adjoint functions. When recursion relations cannot be developed for the adjoint functions, the collocation method is probably more efficient than the spectral expansion method, since the sources for the corresponding adjoint systems are just Dirac delta functions (which makes the respective computation equivalent to the computation of a Green’s function), rather than the more elaborated sources involving high-order Fourier basis functions or orthogonal polynomials. For systems involving many independent variables, it is likely that a hybrid combination of spectral expansions in some independent variables and collocation in the remaining independent variables would provide the most efficient computational outcome.

Turnpike properties of optimal relaxed control problems

In this paper, three kinds of turnpike properties for optimal relaxed control problems are considered. Under some convexity and controllability assumptions, we obtain the uniform boundedness of the optimal pairs and the adjoint functions. On the basis, we prove the integral turnpike property, the mean square turnpike property and the exponential turnpike property, respectively.

The nonlocal boundary problem with perturbations of antiperiodicity conditions for the eliptic equation with constant coefficients

In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$ Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. Also we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G)$. In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$

EIGENVALUE PROBLEM FOR THE LAPLACE OPERATOR WITH DISPLACEMENT IN DERIVATIVES

The statement of the problem on the determination of eigen-and adjointfunctions for Laplace operator in s-dimensional unit ball with displacement in derivatives is given. For s =2 the conditions are obtained for the existence of adjoint functions of the not higher than three order and their computation is made. The case of arbitrary s is the subject of future work.