卷 293, 编号 Suppl 1 (2016)

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’s generative grammar for the language of terms of Taylor’s formula.

A strongly α-uniform partial line space of order (s, t) is called an α-partial geometry. If α = t+1, then the geometry is a dual 2-design. Locally triangular and locally Grassman graphs correspond to triangular extensions of certain dual 2-designs, and the class of strongly uniform quasi-biplanes coincides with the class of strongly uniform extensions of dual 2-designs. We study strongly uniform extensions of dual 2-designs.

Let G be a finite group G, and let N(G) be the set of sizes of its conjugacy classes. It is shown that, if N(G) equals N(Alt_n) or N(Sym_n), where n > 1361, then G has a composition factor isomorphic to an alternating group Altm with m ≤ n and the interval (m, n] contains no primes.

Phase locking is studied in a one-dimensional medium under the action of an external force with slowly changing frequency. In a typical situation, the phase locking is described by a nonstationary nonlinear Schr¨odinger equation with external force. For large values of the time variable, the leading term of a space-localized growing asymptotic solution with soliton profile in the main order is constructed. It turned out that a time-growing asymptotic solution can be obtained for an external excitation with decreasing amplitude. Necessary growth conditions are deduced for such a solution in the presence of dissipation.

The research is focused on the question of proportional development in economic growth modeling. A multilevel dynamic optimization model is developed for the construction of balanced proportions for production factors and investments in a situation of changing prices. At the first level, models with production functions of different types are examined within the classical static optimization approach. It is shown that all these models possess the property of proportionality: in the solution of product maximization and cost minimization problems, production factor levels are directly proportional to each other with coefficients of proportionality depending on prices and elasticities of production functions. At the second level, proportional solutions of the first level are transferred to an economic growth model to solve the problem of dynamic optimization for the investments in production factors. Due to proportionality conditions and the homogeneity condition of degree 1 for the macroeconomic production functions, the original nonlinear dynamics is converted to a linear system of differential equations that describe the dynamics of production factors. In the conversion, all peculiarities of the nonlinear model are hidden in a time-dependent scale factor (total factor productivity) of the linear model, which is determined by proportions between prices and elasticities of the production functions. For a control problem with linear dynamics, analytic formulas are obtained for optimal development trajectories within the Pontryagin maximum principle for statements with finite and infinite horizons. It is shown that solutions of these two problems differ crucially from each other: in finite horizon problems the optimal investment strategy inevitably has the zero regime at the final stage, whereas the infinite horizon problem always has a strictly positive solution. A remarkable result of the proposed model consists in constructive analytical solutions for optimal investments in production factors, which depend on the price dynamics and other economic parameters such as elasticities of production functions, total factor productivity, and depreciation factors. This feature serves as a background for the productive fusion of optimization models for investments in production factors in the framework of a multilevel structure and provides a solid basis for constructing optimal trajectories of economic development.

A problem of tracking a solution of a second-order differential equation in a Hilbert space by a solution of another equation is considered. It is assumed that the first (reference) equation is subject to the action of an unknown control, which is unbounded in time. In the case when the current states of both equations are observed with small errors, a solution algorithm stable with respect to informational noises and computational inaccuracies is designed. The algorithm is based on N.N. Krasovskii’s extremal shift method known in the theory of guaranteed control.

The problem of damping a system of linear oscillators is considered. The problem is solved by using a control in the form of dry friction. The motion of the system under the control is governed by a system of differential equations with discontinuous right-hand side. A uniqueness and continuity theorem is proved for the phase flow of this system. Thus, the control in the form of generalized dry friction defines the motion of the system of oscillators uniquely.

We consider Pontryagin’s generalized nonstationary example with identical dynamic and inertial capabilities of the players under phase constraints on the evader’s states. The boundary of the phase constraints is not a “death line” for the evader. The set of admissible controls is a ball centered at the origin, and the terminal sets are the origin. We obtain sufficient conditions for a multiple capture of one evader by a group of pursuers in the case when some functions corresponding to the initial data and to the parameters of the game are recurrent.

An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation and a perturbation whose values are extremal points of the convexified perturbation that also belong to the values of the original perturbation. Questions of the existence of solutions under various perturbations are studied and relations between solutions are established. The primary focus is on the weakening of assumptions on the perturbation as compared to the known assumptions under which existence and relaxation theorems are valid. All our assumptions, in contrast to the known assumptions, concern the convexified rather than original perturbation.

The problem of an optimal cover of sets in three-dimensional Euclidian space by the union of a fixed number of equal balls, where the optimality criterion is the radius of the balls, is studied. Analytical and numerical algorithms based on the division of a set into Dirichlet domains and finding their Chebyshev centers are suggested for this problem. Stochastic iterative procedures are used. Bounds for the asymptotics of the radii of the balls as their number tends to infinity are obtained. The simulation of several examples is performed and their visualization is presented.

We consider the game problem of approach for a system whose dynamics is described by a differential operator equation in a Hilbert space. The equation is written in an implicit form with generally noninvertible operator multiplying the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in a right half-plane. Using the method of resolving functionals, we obtain sufficient conditions for the approach of a dynamical vector of the system to a cylindrical terminal set. Applications to systems described by partial differential equations are considered.