Том 305, № Suppl 1 (2019)

Let Γ be a distance-regular graph of diameter 3 containing a maximal 1-code C, which is locally regular and last subconstituent perfect. Then the graph Γ has intersection array {a(p + 1), cp, a + 1; 1,c, ap} or {a(p + 1), (a + 1)p, c;1, c, ap}, where a = a₃, c = c₂, and p = p₃₃³ (Jurišić, Vidali). In the first case, Γ has eigenvalue θ₂ = −1 and the graph Γ₃ is pseudogeometric for GQ(p + 1, a). In the second case, Γ is a Shilla graph. We study Shilla graphs in which every two vertices at distance 2 belong to a maximal 1-code. It is proved that, in the case θ₂ = −1, a graph with the specified property is either the Hamming graph H(3, 3) or a Johnson graph. We find necessary conditions for the existence of Q-polynomial Shilla graphs in which any two vertices at distance 3 lie in a maximal 1-code. In particular, we find two infinite families of feasible intersection arrays of Q-polynomial graphs with the specified property: {b(b² − 3b)/2, (b − 2)(b − 1)²/2, (b − 2)t/2; 1, bt/2, (b² − 3b)(b − 1)/2} (graphs with p₃₃³ = 0) and {b²(b − 4)/2, (b² − 4b + 2)(b − 1)/2, (b − 2)l/2; 1, bl/2, (b² − 4b)(b − 1)/2} (graphs with p₃₃³ = 1).

We consider the Laplace operator in an infinite planar strip with a periodic delta interaction. The width of the strip is fixed and for simplicity is chosen equal to π. The delta interaction is introduced on a periodic system of curves. Each curve consists of a finite number of segments, each having smoothness C¹. The curves are supposed to be strictly internal and do not intersect the boundaries of the strip. The period of their location is 2επ, where ε is a sufficiently small number. The function describing the delta interaction is also periodic on the system of curves and is assumed to be bounded and measurable. The main result is the following fact. If ε ≤ ε₀, where ε₀ is a certain explicitly calculated number and the norm of the function describing the delta interaction is smaller than some explicit constant, then the lower part of the spectrum of the operator has no internal gaps. The lower part is understood as the band of the spectrum until some point, which is explicitly calculated in terms of the parameter ε as a rather simple function. This result can be considered as a first step to the proof of the strong Bethe-Sommerfeld conjecture on the complete absence of gaps in the spectrum of an operator for a sufficiently small period of location of delta interactions.

The program iteration method, which is inseparably associated with the name of A. G. Chentsov, first appeared in the study of the so-called zero-sum differential games. At early stages, only one of the two possible dual iterative procedures was considered — the maximin procedure. This can be explained by the special interest of the researchers in the so-called program maximin function, which is conveniently interpreted in geometric terms in games of pursuit. Nevertheless, the dual minimax iterative procedure is of no less interest. The program iteration method is mainly significant because it may be used as the basis for the development of a differential game theory in a closed compact form, which was shown earlier for a version of the method based on a certain modification of iterative operators. The key role in this theory belongs to the theorem that states the existence and uniqueness of a solution of the equation induced by a pair of such operators. In this case, the maximin iterative procedure is used to describe ε-optimal (in some cases, optimal) positional strategies of the first player, while the minimax procedure is used to describe ε-optimal (in some cases, optimal) positional strategies of the second player. This paper investigates the structure of the solution set of the generalized Bellman-Isaacs equation obtained with the use of historically first (not modified) operators of the program iteration method. A theorem that states the existence and uniqueness of the solution to this equation meeting a natural boundary condition is proved under certain assumptions. Thus, it is shown that the original version of the program iteration method can also be used in designing a closed-form differential game theory. However, here we use the so-called recursive strategies rather than positional ones. Such strategies, together with the program iteration method, play an essential role in the analysis of coalition-free differential games.

Functional-differential equations ẋ = f (t, ϕ(·)) with piecewise continuous right-hand sides are studied. It is assumed that the set M of discontinuity points of the right-hand side has empty interior in contrast to being a measure zero sets, as in the case of differential equations without delay. This assumption is made largely because the domain of the function f is infinite-dimensional. Solutions to the equations under consideration are understood in A. F. Filippov’s sense. The main results are theorems on the asymptotic behavior of solutions formulated with the use of invariantly differentiable Lyapunov functionals with constant-sign derivatives. Nonautonomous systems are difficult to deal with because ω-limit sets of their solutions do not possess invariance-type properties, whereas sets of zeros of derivatives of Lyapunov functionals may depend on the variable t and extend beyond the space of variables ϕ(·). For discontinuous nonautonomous systems, there arises the issue of constructing the limiting differential equations with the use of shifts f^τ(t + τ, ϕ(·)) of the function f. We introduce the notion of limiting differential inclusion without employing limit passages on sequences of shifts of discontinuous or multivalued mappings. The properties of such inclusions are studied. Invariance-type properties of ω-limit sets of solutions and analogs of LaSalle’s invariance principle are established.

For functionals defined on variable Sobolev spaces, we establish a series of results on the convergence of minimizers and minimum values on sets of functions subject to implicit constraints by means of periodic rapidly oscillating functions. In connection with the formulation and justification of these results, we introduce the definition of Γ-convergence of functionals corresponding to the given sets of constraints. The specificity of the introduced definition is that it refers to the convergence of a sequence of functionals defined on variable Sobolev spaces to a function on the real line. The considered minimization problems have the feature that, to justify the convergence of a sequence of their solutions, the strong connectedness of the domains of the corresponding functionals is not required, while this connectedness is essential, for instance, in the study of the convergence of solutions of the Neumann variational problems and variational problems with explicit unilateral and bilateral constraints in variable domains. In addition to the mentioned results, we establish theorems on the Γ-compactness of sequences of functionals with respect to the given sets of constraints.

A linear functional differential control system of general form with aftereffect is considered. An optimal control problem with linear constraints on the state and control variables is studied. The control is realized by a linear operator of general form. The cases of distributed and lumped delay in the control loop, as well as the case of impulsive control, are covered. The Cauchy matrix is used to reduce the problem under consideration to a problem formulated only in terms of control variables with the use of some auxiliary variables linked with the defining relations for the Cauchy matrix of the system. In the case where the control is chosen from a finite-dimensional subspace of the control space, a problem effectively solvable by standard software tools is written explicitly. An example of an applied optimal control problem that arises in economic dynamics is presented. A class of hybrid systems (systems with continuous and discrete times) reducible to the system under consideration is described.

In our previous papers (2002, 2017), we derived conditions for the existence of a strong Nash equilibrium in multistage nonzero-sum games under additional constraints on the possible deviations of coalitions from their agreed-upon strategies. These constraints allowed only one-time simultaneous deviations of all the players in a coalition. However, it is clear that in real-world problems the deviations of different members of a coalition may occur at different times (at different stages of the game), which makes the punishment strategy approach proposed by the authors earlier inapplicable in the general case. The fundamental difficulty is that in the general case the players who must punish the deviating coalition know neither the members of this coalition nor the times when each player performs the deviation. In this paper, we propose a new punishment strategy, which does not require the full information about the deviating coalition but uses only the fact of deviation of at least one player of the coalition. Of course, this punishment strategy can be realized only under some additional constraints on stage games. Under these additional constraints, it was proved that the punishment of the deviating coalition can be effectively realized. As a result, the existence of a strong Nash equilibrium in the game was established.

We study the stability properties of generalized local splines of the form

\(S(x) = S(f,x) = \sum\limits_{j \in \mathbb{Z}} {{y_j}{B_\varphi }\left( {x + \frac{{3h}}{2} - jh} \right),}\;\;\; x \in \mathbb{R},\)

\({B_\varphi }(x) = m(h)\left\{ {\begin{array}{*{20}{c}} {\varphi (x),}&{x \in [0;h],} \\ {2\varphi (h) - \varphi (x - h) - \varphi (2h - x),}&{x \in [h;2h],} \\ {\varphi (3h - x),}&{x \in [2h;3h],} \\ {0,}&{x \notin [0;3h],} \end{array}} \right.\)

We derive estimates for the Hausdorff distance between sets and their convex hulls in finite-dimensional Euclidean spaces with the standard inner product and the corresponding norm. In the first part of the paper, we consider estimates for α-sets. By an α-set we mean an arbitrary compact set for which the parameter characterizing the degree of nonconvexity and computed in a certain way equals α. In most cases, the parameter α is the maximum possible angle under which the projections to this set of points not belonging to the set are visible from these points. Note that α-sets were introduced by Ushakov for the classification of nonconvex sets according to the degree of their nonconvexity; α-sets are used for the description of wavefronts and for the solution of other problems in control theory. We consider α-sets only in a two-dimensional space. It is proved that, if α is small, then the corresponding α-sets are close to convex sets in the Hausdorff metric. This allows us to neglect their nonconvexity and consider such sets convex if it is known that the parameter α is small. The Shapley-Folkman theorem is often applied in the same way. In the second part of the paper, we present an improvement of the estimate from the Shapley-Folkman theorem. The original Shapley-Folkman theorem states that the Minkowski sum of a large number of sets is close in the Hausdorff metric to the convex hull of this sum with respect to the value of the Chebyshev radius of the sum. We consider a particular case when the sum consists of identical terms; i.e., we add some set M to itself. For this case, we derive an improved estimate, which is essential for sets in spaces of small dimension. In addition, as in Starr’s known corollary, the new estimate admits the following improvement: the Chebyshev radius R(M) on the right-hand side can be replaced by the inner radius r(M) of the set M. However, as the dimension of the space grows, the new estimate tends asymptotically to the estimate following immediately from the Shapley-Folkman theorem.