Vol 304, No Suppl 1 (2019)

We consider an optimal control problem for an autonomous differential inclusion with free terminal time and a mixed functional which contains the characteristic function of a given open set M ⊂ ℝⁿ in the integral term. The statement of the problem weakens the statement of the classical optimal control problem with state constraints to the case where the presence of admissible trajectories of the system in the set M is physically allowed but undesirable due to safety or instability reasons. Using an approximation approach, necessary conditions for the optimality of an admissible trajectory are obtained in the form of Clarke’s Hamiltonian inclusion. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of the problem with a state constraint, the obtained necessary optimality conditions may degenerate. Conditions guaranteeing their nondegeneracy and pointwise nontriviality are presented. The results obtained extend the author’s previous results to the case of a problem with free terminal time and more general functional.

We study groups of classical links, welded links, and virtual links. For classical braids, it is proved that the closures of a braid and its automorphic image are weakly equivalent. This implies the affirmative answer to the question of the coincidence of the groups constructed from a braid and from its automorphic image. We also study the problem of residual nilpotence of groups of virtual knots. It is known that, in a classical knot group, the commutator subgroup coincides with the third term of the lower central series and, hence, the quotient by the terms of the lower central series yields nothing. We prove that the situation is different for virtual knots. A nontrivial homomorphism of the virtual trefoil group to a nilpotent group of class 4 is constructed. We use the Magnus representation of a free group by power series to construct a homomorphism of the virtual trefoil group to a finite-dimensional algebra. This produces the nontrivial linear representation of the virtual trefoil group by unitriangular matrices of order 8.

A conflict-controlled process of the approach of a trajectory to a cylindrical terminal set is studied. The problem statement encompasses a wide range of quasilinear functional-differential systems. We use the technique of set-valued mappings and their selections to derive sufficient conditions for the game termination in a finite time. The methodology used is close to the scheme that involves the time of the first absorption. By way of illustration, quasilinear integro-differential games are examined. For this purpose, their solutions are presented in the form of an analog of the Cauchy formula. The calculations are performed for the case of a system with a simple matrix; the control sets of the players are balls centered at the origin and the terminal set is a linear subspace. Depending on the relations between the initial state of the system and the parameters of the process, sufficient conditions for the game termination are derived. An explicit form of the guaranteed time is found in one specific case.

A minimax solution of the Cauchy problem for a functional Hamilton-Jacobi equation with coinvariant derivatives and a condition at the right end is considered. Hamilton-Jacobi equations of this type arise in dynamical optimization problems for time-delay systems. Their approximation is associated with additional issues of a valid transition from an infinite-dimensional functional argument of the desired solution to a finite-dimensional argument. Earlier, the schemes based on the piecewise linear approximation of the functional argument and the correctness properties of minimax solutions were studied. In this paper, a scheme for the approximation of Hamilton-Jacobi functional equations with coinvariant derivatives by usual Hamilton-Jacobi partial differential equations is proposed and justified. The scheme is based on the approximation of the characteristic functional-differential inclusions used in the definition of the desired minimax solution by ordinary differential inclusions.

The Steiner problem is considered in the Gromov-Hausdorff space, i.e., in the space of compact metric spaces (considered up to isometry) endowed with the Gromov-Hausdorff distance. Since this space is not boundedly compact, the problem of the existence of a shortest network in this space is open. It is shown that each finite family of finite metric spaces can be connected by a shortest network. Moreover, it turns out that in this case there exists a shortest tree all of whose vertices are finite metric spaces. An estimate for the number of points in these metric spaces is obtained. As an example, the case of three-point metric spaces is considered. It is also shown that the Gromov-Hausdorff space does not realize minimal fillings; i.e., shortest trees in this space need not be minimal fillings of their boundaries.

A graph Γ is called t-isoregular if, for any i ≤ t and any i-vertex subset S, the number Γ(S) depends only on the isomorphism class of the subgraph induced by S. A graph Γ on v vertices is called absolutely isoregular if it is (v — 1)-isoregular. It is known that each 5-isoregular graph is absolutely isoregular, and such graphs have been fully described. Each precisely 4-isoregular graph is either a pseudogeometric graph for pG_r(2r, 2r³ + 3r² — 1) or its complement. A pseudogeometric graph for pG_r(2r, 2r³ + 3r² — 1) is denoted by Izo(r). Graphs Izo(r) do not exist for an infinite set of values of r (r = 3, 4, 6, 10,…). The existence of Izo(5) is unknown. In this work, we find possible automorphisms for the neighborhood of an edge from Izo(5).

The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine and their followers at the beginning of the century. This theory considers finite graphs as analogs of Riemann surfaces and branched coverings of graphs as holomorphic mappings. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of half-edges of the graph. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus g > 1 does not exceed 84(g — 1). Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a discrete analog of the E. Bujalance and G. Gromadzki theorem improving one of Arakawa’s results is obtained.

Differential equations and control systems with impulse action and random parameters are considered. These objects are characterized by stochastic behavior: the lengths θ_k of the intervals between the times of the impulses τ_k, k = 0,1,…, are random variables and the magnitudes of the impulses also depend on random actions. The basic object of research is the control system

which depends on random parameters θ_k = τ_k+1 — τ_k and v_k, k = 0,1,…. A probability measure μ is defined on the set Σ of all possible sequences ((θ₀, v₀),…, (θ_k, v_k),… ). Admissible controls u = u(t) are bounded measurable functions with values in a compact set U ⊂ R^m, and the vector w_k is also a control affecting the behavior of the system at the times τ_k. We consider the set \(\mathfrak{M}=\{(t,x):t\in[0,+\infty),x\in{M}(t)\}\) defined by the function t ↦ M(t), which is continuous in the Hausdorff metric. The main result of the paper is sufficient conditions for the Lyapunov stability and asymptotic stability of the set M with probability 1. It is shown that the stability analysis of a set by means of the method of Lyapunov functions can be reduced to studying the stability of the zero solution of the corresponding differential equation. We also study the asymptotic behavior of solutions of differential equations with impulse action and random parameters. Conditions are obtained under which the solutions possess the Lyapunov stability and asymptotic stability for all values of the random parameter and with probability 1. The results are illustrated by a probability model of a population subject to harvesting and by a model of two-species competition with impulse action.

We consider a linear differential game in which the first player can choose both an impulse control and a control subject to a geometric constraint. The first player can use a prescribed amount of resource to form the impulse control. Portions of this amount can be separated at certain times, thus producing “instantaneous” changes of the state vector and complicating the problem. The control of the second player is subject to a geometric constraint. The vectograms of the players are described by the same ball with different time-dependent radii. The terminal set is a ball with fixed radius. The aim of the first player is to bring the state vector to the terminal set at a given time. The aim of the second player is opposite. Necessary and sufficient conditions for meeting the terminal set at the given time are found, and the corresponding controls of the players guaranteeing the achievement of their goals are constructed. A solution of an example illustrating the theory is given.

Suppose that G is a finite group, π(G) is the set of prime divisors of its order, and ω(G) is the set of orders of its elements. We define a graph on π(G) with the following adjacency relation: different vertices r and s from π(G) are adjacent if and only if rs ∈ ω(G). This graph is called the Gruenberg-Kegel graph or the prime graph of G and is denoted by GK(G). In a series of papers, we describe the coincidence conditions for the prime graphs of nonisomorphic simple groups. This issue is connected with Vasil’ev’s Question 16.26 in the Kourovka Notebook about the number of nonisomorphic simple groups with the same prime graph. Earlier the author derived necessary and sufficient conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type over fields of orders q and q₁, respectively, with the same characteristic. Let G and G₁ be two nonisomorphic finite simple groups of Lie type over fields of orders q and q₁, respectively, with different characteristics. The author also obtained necessary conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type. In the present paper the latter result is refined in the case where G is a simple linear group of sufficiently high Lie rank over a field of order q. If G is a simple linear group of sufficiently high Lie rank, then we prove that the prime graphs of G and G₁ may coincide only in one of the nineteen cases. As corollaries of the main result, we obtain constraints (under some additional conditions) on the possible number of simple groups whose prime graph is the same as the prime graph of a simple linear group.