Yu. N. Kiselev

Optimal resource allocation program in a two-sector economic model with a Cobb-Douglas production function

We consider the resource allocation problem for a two-sector economic model with a two-factor Cobb-Douglas production function on a finite time horizon with a terminal functional. The problem is reduced to some canonical form by a scaling of the phase variables and time. We prove the optimality of the extremal solution constructed on the basis of the maximum principle. The solution of the boundary value problem of the maximum principle is constructed in closed form for three cases of location of the initial plant state.

Profit maximization problem for Cobb–Douglas and CES production functions

Production functions are used to model the production activity of enterprises. In this article, we formulate the necessary and sufficient conditions of strict concavity for Cobb–Douglas and constant elasticity of substitution (CES) production functions. These conditions constitute the theoretical foundation for analyzing the profit maximization problem. An optimal solution is constructed in analytical form and some of its properties are described. Three approaches to solving the profit maximization problem are considered and their equivalence is established. For a Cobb–Douglas production function we investigate the dependence of the maximum profit on elasticity coefficients. A similar analysis is carried out also for the CES production function. The article presents a systematic and detailed discussion of the relevant topics. The topic is related to the investigation of innovation activity of enterprises. The theoretical results and the explicit analytical relationships provide a theoretical and algorithmic base for the “Planer” optimization software—a useful product for the analysis of the production activity of enterprises modeled by production function tools.

Some algorithms of optimal control

In the first part of the paper, we describe the method of continuation with respect to a parameter in solution algorithms for nonlinear boundary value problems in ordinary differential equations. We present results of numerical experiments solving boundary value problems, including boundary value problems arising in optimal control theory. The parameter variation scheme (the continuation method) can be considered as a special development and modification of the classical Newton method. The basic idea of this approach can be shortly formulated as reducing a boundary value problem to a Cauchy problem. Regarding a Cauchy problem as an elementary operation, we arrive at a compact description of the algorithm of solving a boundary value problem by means of the method of continuation with respect to a parameter. The interest in this research area is related to studying numerical algorithms of solving the linear time-optimal control problem and is aimed at boundary problems of the maximum principle. We have developed a program BVP, which solves in the Maple environment regular boundary value problems for ordinary differential equations, some boundary value problems of the maximum principle arising in optimal control, problems of finding periodic solutions and limit cycles, and so on. In the second part of the paper, we describe a simple algorithm of constructing attainability (controllability) sets in plane linear controlled systems and give some examples of using it. The algorithm is based on parametric equations of the boundary of a plane strictly convex compact set given by its support function. This approach allows one to construct two-dimensional projections of attainability sets for multidimensional linear controlled systems. In the third part of the paper, we present sufficient optimality conditions for nonlinear controlled systems in terms of constructions of the Pontryagin maximum principle.

Optimal Resource Distribution Program in a Two-Sector Economic Model with a Cobb-Douglas Production Function with Distinct Amortization Factors

We consider a resource distribution problem on a finite time interval with a terminal functional for a two-sector economic model with a two-factor Cobb-Douglas production function with distinct amortization factors. The problem can be reduced to a canonical form by scaling the state variables and time. We prove the optimality of an extremal solution constructed with the use of the maximum principle. For the case in which the initial state of the plant lies above the singular ray, the solution of the boundary value problem of the maximum principle is presented in closed form.

Resource allocation problem in a two-sector economic model of special form

In the present paper, we study the resource allocation problem for a two-sector economic model of special form, which is of interest in applications. The optimization problem is considered on a given finite time interval. We show that, under certain conditions on the model parameters, the optimal solution contains a singular mode. We construct optimal solutions in closed form. The theoretical basis for the obtained results is provided by necessary optimality conditions (the Pontryagin maximum principle) and sufficient optimality conditions in terms of constructions of the Pontryagin maximum principle.

Optimal resource allocation program in a two-sector economic model with an integral type functional for various amortization factors

We study the resource allocation problem in a two-sector economic model with a two-factor Cobb–Douglas production function for various amortization factors on a finite time horizon with a functional of the integral type. The problem is reduced to a canonical form by scaling the state variables and time. We show that the extremal solution constructed with the use of the maximum principle is optimal. For a sufficiently large planning horizon, the optimal control has two or three switching points, contains one singular segment, and is zero on the terminal part. The considered problem with different production functions admits a biological interpretation in a model of balanced growth of plants on a given finite time interval.

Boundary value problem of Pontryagin’s maximum principle in a two-sector economy model with an integral utility function

An infinite-horizon two-sector economy model with a Cobb–Douglas production function and a utility function that is an integral functional with discounting and a logarithmic integrand is investigated. The application of Pontryagin’s maximum principle yields a boundary value problem with special conditions at infinity. The search for the solution of the maximum-principle boundary value problem is complicated by singular modes in its optimal solution. In the construction of the solution to the problem, they are described in analytical form. Additionally, a special version of the sweep method in continuous form is proposed, which is of interest from theoretical and computational points of view. An important result is the proof of the optimality of the extremal solution obtained by applying the maximum-principle boundary value problem.

Investigating a two-sector model with an integrating-type functional cost

A resource allocation problem in a two-sector model with the Cobb-Douglas production function and integrating-type functional cost with discounting is considered. The planning horizon is finite, fixed, and quite large. A constructive description of the optimum solution is proposed. The solution is based on the Pontryagin maximum principle. The optimality of an extreme solution is proved using the theorem on the sufficient optimality conditions in terms of constructions of the maximum principles. The studied problem with different production functions is open to biological interpretation under the model of balanced growth of plants within a given limited time span.

Studying Mathematical Models of Resources Allocation Among a Cell’s Assimilator Mechanisms

Microbes assimilate nutrients, getting building material from them. This material is spent in the process of growth. Building material is allocated among different types of macromolecular structures. Thus, the resource allocation problem arises. In this paper, we consider one of the aspects of this problem, namely, the resource allocation among various molecular mechanisms that are involved in the assimilation and processing of various nutrients. Another aspect, the problem of optimal allocation of a microorganism colony’s resources between growth and nutrient assimilation processes was studied in [7]. The resource allocation problem in the processes of nutrient assimilation and processing is of great interest because the allocating mechanism determines the rate of corresponding nutrient intake by a cell; on the other hand, the proportion between nutrients in a cell can be used to work out the corresponding allocation mode. In order for the mathematical model to be as simple as possible, we assume that a microbe’s cell consists of only two chemical elements, carbon and nitrogen. A cell is assumed to be capable of synthesizing only two types of elements of the nutrient assimilation mechanism. One of them assimilates nutrients from which a cell gets carbon (and only carbon); the other assimilates nutrients from which this cell gets nitrogen (and only nitrogen). Thus, the problem of allocating the building material between the assimilation of carbon-containing nutrients and that of nitrogen-containing nutrients arises. Of course, the assumptions formulated above simplify the actual situation quite substantially. However, we believe that conclusions obtained on the basis of the analysis of this simplified situation remain valid in more complex cases, which are described in the literature (see, e.g., [6]). In addition, it is assumed that carbon and nitrogen are needed in fixed proportion for the synthesis of the molecular mechanism and that the cell uses all building material to build up the nutrient assimilation mechanism. Then, the following three cases are possible:

Analysis of Trajectories of a Nonlinear Systems of Differential Equations

The resource allocation problem in the assimilation and processing of nutrients is of interest, since the allocation mechanism determines the uptake rate of the corresponding nutrients into a cell; on the other hand, in turn, the ratio of nutrients in a cell can be used in forming the related allocation mode. To simplify the mathematical model as much as possible, we assume that a microbe cell consists only of two chemical elements, carbon and nitrogen. We also assume that the cell can synthesize only two types of elements of assimilatory machinery, one assimilating nutrients from which the cell gets carbon (and only carbon), and the other assimilating nutrients from which the cell gets nitrogen (and only nitrogen). Therefore, we face the problem of allocating the building material to the carbon- and nitrogen-assimilatory machinery. In addition, we assume that the synthesis of the machinery requires given proportions of carbon and nitrogen and the cell uses all building material for the growth of the assimilatory machinery. Then the following three cases are possible: