Proceedings of the Steklov Institute of Mathematics最新文献

A quadratic minimization problem is considered in Hilbert spaces under constraints given by a linear operator equation and a convex quadratic inequality. The main feature of the problem statement is that the practically available approximations to the exact linear operators specifying the criterion and the constraints converge to them only strongly pointwise rather than in the uniform operator norm, which makes it impossible to justify the use of the classical regularization methods. We propose a regularization method that is applicable in the presence of error estimates for approximate operators in pairs of other operator norms, which are weaker than the original ones. For each of the operators, the pair of corresponding weakened operator norms is obtained by strengthening the norm in the domain of the operator and weakening the norm in its range. The weakening of operator norms usually makes it possible to estimate errors in operators where this was fundamentally impossible in the original norms, for example, in the finite-dimensional approximation of a noncompact operator. From the original optimization formulation, a transition is made to the problem of finding a saddle point of the Lagrange function. The proposed numerical method for finding a saddle point is an iterative regularized extragradient two-stage procedure. At the first stage of each iteration, an approximation to the optimal value of the criterion is refined; at the second stage, the approximate solution with respect to the main variable is refined. Compared to the methods previously developed by the authors and working under similar information conditions, this method is preferable for practical implementation, since it does not require the gradient step size to converge to zero. The main result of the work is the proof of the strong convergence of the approximations generated by the method to one of the exact solutions to the original problem in the norm of the original space.

The paper considers minimization problems with a free right endpoint on a given time interval for control affine systems of differential equations. For this class of problems, we study an estimate for the number of different zeros of switching functions that determine the form of the corresponding optimal controls. This study is based on analyzing nonautonomous linear systems of differential equations for switching functions and the corresponding auxiliary functions. Nonautonomous linear systems of third order are considered in detail. In these systems, the variables are changed so that the matrix of the system is transformed into a special upper triangular form. As a result, the number of zeros of the corresponding switching functions is estimated using the generalized Rolle’s theorem. In the case of a linear system of third order, this transformation is carried out using functions that satisfy a nonautonomous system of quadratic differential equations of the same order. The paper presents two approaches that ensure the extensibility of solutions of a nonautonomous system of quadratic differential equations to a given time interval. The first approach uses differential inequalities and Chaplygin’s comparison theorem. The second approach combines splitting a nonautonomous system of quadratic differential equations into subsystems of lower order and applying the quasi-positivity condition to these subsystems.

A subgroup (H) of a group (G) is pronormal if, for each (gin G), the subgroups (H) and (H^{g}) are conjugate in (langle H,H^{g}rangle). Most of finite simple groups possess the following property ((ast)): each subgroup of odd index is pronormal in the group. The conjecture that all finite simple groups possess the property ((ast)) was established in 2012 in a paper by E.P. Vdovin and the third author based on the analysis of the proof that Hall subgroups are pronormal in finite simple groups. However, the conjecture was disproved in 2016 by A.S. Kondrat’ev together with the second and third authors. In a series of papers by Kondrat’ev and the authors published from 2015 to 2020, the finite simple groups with the property ((ast)) except finite simple linear and unitary groups with some constraints on natural arithmetic parameters were classified. In this paper, we construct series of examples of nonpronormal subgroups of odd index in finite simple linear and unitary groups over a field of odd characteristic, thereby making a step towards completing the classification of finite simple groups with the property ((ast)).

A traditional object of study in the mathematical theory of optimal control is a controlled object with geometric constraints on the control vector (u). At the same time, it turns out that sometimes it is more convenient to impose integral constraints on the control vector (u). For example, in the theory of automatic design of optimal controllers, it is sometimes assumed that the control vector (u) is not subject to any geometric constraints, but there is a requirement that the control (u(t)) and its squared length (|u(t)|^{2}) are Lebesgue summable on the corresponding interval. This circumstance, as well as the fact that the performance index has the form of a quadratic functional, makes it possible to construct an optimal control under rather broad assumptions. Quadratic integral constraints on controls can be interpreted as some energy constraints. Controlled objects under integral constraints on the controls are given quite a lot of attention in the mathematical literature on control theory. We mention the works of N.N. Krasovskii, E.B. Lee, L. Markus, A.B. Kurzhanskii, M.I. Gusev, I.V. Zykov, and their students. The paper studies a linear time-optimal problem, in which the terminal set is the origin, under an integral constraint on the control. Sufficient conditions are obtained under which the optimal time as a function of the initial state (x_{0}) is continuous.

The issues of existence and uniqueness of a solution to the Cauchy problem are studied for a linear equation in a Banach space with a closed operator at the unknown function that is resolved with respect to a first-order integro-differential operator of Gerasimov type. The properties of resolving families of operators of the homogeneous equations are investigated. It is shown that sectoriality, i.e., belonging to the class of operators (mathcal{A}_{K}) introduced here, is a necessary and sufficient condition for the existence of an analytical resolving family of operators in a sector. A theorem on the perturbation of operators of the class (mathcal{A}_{K}) is obtained, and two versions of the theorem on the existence and uniqueness of a solution to a linear inhomogeneous equation are proved. Abstract results are used to study initial–boundary value problems for an equation with the Prabhakar time derivative and for a system of partial differential equations with Gerasimov–Caputo time derivatives of different orders.

In the framework of Baoding Liu’s uncertainty theory, some new concepts are introduced and their properties are considered.In particular, regular functions of uncertainty are introduced on an uncountable product of spaces. An analog of the Łomnicki–Ulamtheorem from traditional probability theory is obtained. Necessary and sufficient conditions are specified under which a functiondefined on a Banach space of bounded functions is a distribution function for some uncertain mapping. Some notions of Liu’s theory aregeneralized for uncountably many objects. Examples showing the similarity and the difference between Liu’s theory and probability theoryare analyzed. An application of Liu’s theory to estimation theory is considered with examples.

We consider vector bundles of rank (2) with trivial generic fiber on the projective line over (mathbb{Z}). For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to (mathcal{O}^{2}) in the fiber over (mathbb{Q}) and are isomorphic to (mathcal{O}^{2}) or (mathcal{O}(-1)oplusmathcal{O}(1)) over each closed point of Spec((mathbb{Z})), we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.

The problem of calculating points and magnitudes of discontinuities in the controls acting on a systemdescribed by a nonlinear vector ordinary differential equation is considered. A similar problem is well known insystems theory and belongs to the class of failure identification problems. This paper specifies a regularizing algorithmthat solves the problem synchronously with the process of functioning of the control system.The algorithm is based on a feedback control method called the dynamic regularization method in the literature;this method was previously actively used in problems of online reconstruction of nonsmooth unknown disturbances.The algorithm described in this work is stable to information noises and computational errors.

Abstract

A mathematical generalization is given of the famous Bell inequality, which arose in connection with the analysis of the classical Einstein–Podolsky–Rosen paradox.

Abstract

General methods of local continuity analysis of characteristics of infinite-dimensional composite quantum systems are considered. A new approximation technique for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This technique is used to prove several general results (a Simon-type dominated convergence theorem, a theorem on the preservation of continuity under convex mixtures, etc.). Local continuity conditions are derived for the following characteristics of composite quantum systems: the quantum conditional entropy, the quantum (conditional) mutual information, the one-way classical correlation and its regularization, the quantum discord and its regularization, the entanglement of formation and its regularization, and the constrained Holevo capacity of a partial trace and its regularization.