S. M. Orlov

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.

Investigating One Class of Extended Open-Loop Guidance Problems

An alternative solvability criterion and guiding control construction procedure are proposed in one class of extended open-loop guidance problems. A control system is described by linear differential equations. The control domain is a rectangular parallelepiped, and the convex and closed target set has a special form. An equivalent system of linear inequalities is constructed for the extended open-loop guidance problem, and a correspondence between their solutions is established. The difference between the developed scheme and the known research techniques of this class of problems is the possibility of solvability criterion verification and search for guiding control in analytical form in a finite number of steps. The proposed technique for solving the extended openloop guidance problem is illustrated using a concrete example.

Algorithm for Constructing a Guaranteeing Program Package in a Control Problem with Incomplete Information

A package control problem is considered for a target set at a moment of time. The dynamic system under control is described by linear differential equations, the control area is a convex compact, and the target set is convex and closed. A version of the subsequent approximations method in extended space is proposed for constructing elements of a guaranteeing program package in the case of regular clusters.

Optimal modes in a multidimensional model of economic growth

An n-dimensional problem of optimal economic growth in a multifactor model with the Cobb–Douglas production function and an integral-type functional with discounting is investigated. The model is studied by assuming that all amortization coefficients are equal. A constructive description of an optimal solution for a sufficiently large planning horizon and a sufficiently small discount coefficient is obtained. The extremal solution is described in analytical form. The studied problem with other production functions has a biological interpretation in an optimal growth model of agricultural plants with n vegetative organs during a specific finite time interval.

Optimal processes in the model of two-sector economy with an integral utility function

An infinite-horizon two-sector economy model with a Cobb–Douglas production function is studied for different depreciation rates, the utility function being an integral functional with discounting and a logarithmic integrand. The application of the Pontryagin maximum principle leads to a boundary value problem with special conditions at infinity. The presence of singular modes in the optimal solution complicates the search for a solution to the boundary value problem of the maximum principle. To construct the solution to the boundary value problem, the singular modes are written in an analytical form; in addition, a special version of the sweep algorithm in continuous form is proposed. The optimality of the extremal solution is proved.

Special modes in a two-sector economy model with an integral utility function

In this work, we study a two-sector economic model with the Cobb–Douglas production function on an infinite planning horizon where the utility function is a functional of an integral form and a Lagrangian of a logarithmic type. A one-dimensional equation is obtained that depends only on the coefficients of elasticity and amortization, and determines the possible special modes. The special modes are described in analytical form.

Studying a modified ramsey model with variable production flexibility

A one-dimensional nonlinear problem of optimal control on an infinite planning horizon is considered that is a modification of Ramsey‘s model for endogenous economic growth with the Cobb–Douglas production function. The model is innovative in its consideration of variable production flexibility, a parameter of the Cobb–Douglas function. As the first step in the study, the problem is considered for piecewise-smooth and piecewise-constant flexibility functions. Optimal solutions are constructed on the basis of a special integral representation of the functional and are unique for each type of flexibility function. The optimal solutions include singular modes. The final results can be used to estimate the effect of uncertainty in a Ramsey problem with constant parameters.