P.D. Lebedev

Construction of a minimax solution for an eikonal-type equation

A formula for a minimax (generalized) solution of the Cauchy-Dirichlet problem for an eikonal-type equation is proved in the case of an isotropic medium providing that the edge set is closed; the boundary of the edge set can be nonsmooth. A technique of constructing a minimax solution is proposed that uses methods from the theory of singularities of differentiable mappings. The notion of a bisector, which is a representative of symmetry sets, is introduced. Special points of the set boundary—pseudovertices—are singled out and bisector branches corresponding to them are constructed; the solution suffers a “gradient catastrophe” on these branches. Having constructed the bisector, one can generate the evolution of wave fronts in smoothness domains of the generalized solution. The relation of the problem under consideration to one class of time-optimal dynamic control problems is shown. The efficiency of the developed approach is illustrated by examples of analytical and numerical construction of minimax solutions.

Procedures for Calculating the Nonconvexity Measures of a Plane Set

The geometry of nonconvex sets is analyzed. The measure of nonconvexity of a closed set that has the sense of an angle is considered. Characteristic manifolds of nonconvex sets are constructed. Procedures for calculating the measure of nonconvexity are proposed for a class of plane sets.

Some Facts about the Ramsey Model

In modeling the dynamics of capital, the Ramsey equation coupled with the Cobb–Douglas production function is reduced to a linear differential equation by means of the Bernoulli substitution. This equation is used in the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. This result enriches our understanding of the model analysis in the optimal control framework.

Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints

We consider a neoclassical (economic) growth model. A nonlinear Ramsey equation, modeling capital dynamics, in the case of Cobb-Douglas production function is reduced to the linear differential equation via a Bernoulli substitution. This considerably facilitates the search for a solution to the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. A proposed algorithm for constructing growth trajectories combines methods of open-loop control and closed-loop regulatory control. For some levels of constraints and initial conditions, a closed-form solution is obtained. We also demonstrate the impact of technological change on the economic equilibrium dynamics. Results are supported by computer calculations.

Comparison of the Positional Absorption Sets in Two Game Approach Problems

Abstract For a given compact target set M ⊂ R n the problem of approach at a given time instant θ and the problem of approach to a given time instant θ are discussed. The coincidence criteria of positional absorbtion sets, i.e. the maximal u -stable bridges are obtained.

Symmetry Sets in Construction of a Minimax Solution for a Bellman-Isaacs Equation

Abstract A formula for a minimax (generalized) solution of the Cauchy-Dirichlet problem for a Bellman–Isaacs equation is proved in the case of an isotropic medium providing that the edge set is closed. A technique of constructing a minimax solution is proposed, which uses methods from the theory of singularities of differentiable mappings. The notion of a bisector, which is a representative of symmetry sets, is introduced. Special points of the set boundary—pseudovertices—are singled out and bisector branches corresponding to them are constructed; the solution suffers a “gradient catastrophe” on these branches. Having constructed the bisector, one can generate the evolution of wave fronts in smoothness domains of the generalized solution. The relation of the problem under consideration to one class of time-optimal dynamic control problems is shown. The efficiency of the developed approach is illustrated by examples of analytical and numerical construction of minimax solutions.