An algebraic approach for $${\mathcal{H}}$$-matrix preconditioners

Hierarchical matrices ( -matrices) approximate matrices in a data-sparse way, and the approximate arithmetic for -matrices is almost optimal. In this paper we present an algebraic approach for constructing -matrices which combines multilevel clustering methods with -matrix arithmetic to compute the -inverse, -LU, and the -Cholesky factors of a matrix. Then the -inverse, -LU or -Cholesky factors can be used as preconditioners in iterative methods to solve systems of linear equations. The numerical results show that this method is efficient and greatly speeds up convergence compared to other approaches, such as JOR or AMG, for solving some large, sparse linear systems, and is comparable to other -matrix constructions based on Nested Dissection.     

On partial detectability of the nonlinear dynamic systems

Conditions were obtained under which the uniform stability (uniform asymptotic stability) in one part of the variables of the zero equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position relative to another, larger part of variables. Conditions were also obtained under which the uniform stability (uniform asymptotic stability) in one part of variables of the “partial” (zero) equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position. These conditions complement a number of the well-known results of the theory of partial stability and partial detectability of the nonlinear dynamic systems. Application of the results obtained to the problems of partial stabilization of the nonlinear control systems was considered.     

On the properties of the identically singular Lagrange problem

We consider the identically singular Lagrange problem of the calculus of variations. It is investigated how the conjugate points and the Jacobi equation index are related to the solvability conditions of the appropriate matrix Riccati equation and the reducibility of the functional to a perfect square. The criterion of the trajectory variation smallness at a small variation of the functional in the neighborhood of the extremal is obtained.     

On polynomial solutions of the linear stationary control system

It is known that the controllable system x′ = Bx + Du, where the x is the n-dimensional vector, can be transferred from an arbitrary initial state x(0) = x ⁰ to an arbitrary finite state x(T) = x ^T by the control function u(t) in the form of the polynomial in degrees t. In this work, the minimum degree of the polynomial is revised: it is equal to 2p + 1, where the number (p ? 1) is a minimum number of matrices in the controllability matrix (Kalman criterion), whose rank is equal to n. A simpler and a more natural algorithm is obtained, which first brings to the discovery of coefficients of a certain polynomial from the system of algebraic equations with the Wronskian and then, with the aid of differentiation, to the construction of functions of state and control.     

On construction of measurement means on the basis of adaptation principle

An idea of the adaptive approach to the construction (design) of measuring systems is developed. An analysis is given of the features of technical adaptive systems and also the specific character of the means of measurements of both their subclass and the object of application of the adaptation principle. On the basis of the analysis, the concept is set up of the construction of these means, according to which the given principle is considered as an instrument of their perfection and its application is extended to all stages of their design from the choice of the principle (principles) of the action to the organization of the procedure of the output control (primary verification) in their series production. The effectiveness of the concept is illustrated by examples of the solution of the earlier unsolved problems and statement of new problems.     

On calculation of the power indices with allowance for the agent preferences

In the general case, complexity of the algorithm to calculate the power indices grows exponentially with the number of voting agents. Yet the volume of calculations may be reduced dramatically if many coalitions have equal numbers of votes. The well-known algorithm for calculation of the Banzhaf and Shapley-Shubik indices was generalized, which enables fast calculation of the power indices where entry of the voting agent into a coalition depends on its preferences over the set of the rest of agents.     

On approximation of the subdifferential of the nonsmooth penalty functional in the problems of optimal control

For the problem of optimal program control with terminal equality constraints which is considered as an “elementary operation” within the framework of the algorithm to calculate the optimal positional control, a smooth approximation the Frechét-nondifferentiable penalty functional was proposed. Examples of numerical experiments were presented.     

On differential equations arising in the theory of processor sharing queues

We give a probabilistic interpretation of some equations arising when the queueing system M/G/1 with egalitarian processor sharing is studied. To get this aim, we use, in addition to the new analytic method of the author, the well-known method of collective marks, in which two separate operations are involved: the “marking” of jobs, and the observation of “catastrophe” processes.     

On the relationship between the problem of anisotropy-based controller synthesis and classical optimal control problems

Consideration is given to the interrelation between different problems of \(\mathcal{H}_2 \)-and \(\mathcal{H}_\infty \)-optimization and their relationship with the theory of anisotropy-based control. It was shown that the anisotropy-based controller of the full order for the completely defined linear system minimizes the mean amount of the transmitted information between the input and output of the closed system.     

On the method of dynamic programming for linear-quadratic problems of optimal control in hybrid systems

For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid linear-quadratic optimal control based on the recently suggested principle of maximum. Interrelations between the hybrid principle of maximum and the method of dynamic programming for the systems of this class were discussed. The classical formalism was extended, the corresponding Riccati equations were obtained, and discontinuity of the “hybrid” Riccati matrix was proved. The computational aspects of the established theoretical results were considered.     

On dual problems of optimal control

The common structure of duality of optimal control problems is suggested, which is based on the extension principle. The iterative computational algorithm is described, in which the primal and the dual problem are solved in parallel.     

On analysis of periodic polling systems

New alternative approach to the analysis of polling systems is developed. It is based on the theory of decomposable semi-regenerative processes.     

On absolute stability of discrete systems

Modernization is considered of the criterion of absolute stability of discrete systems [1, 2], which uses the method of quadratic transformation of the state vector. The modernization aim is to obtain the criterion form that does not explicitly depend on the notions of this method. The new formulation permits performing all calculations in the space of the initial dimension, which reduces the complexity of the problem solution. This work represents the development of the approach suggested in [3] for continuous systems.     

On one problem of integer optimization

A recognition problem of the following form is studied: to find put for the prescribed polyhedron whether the maximum of the linear objective function is achieved at its integral point. It is established that this problem is NP-hard in the general case and polynomially solvable in the class of rooted semimetric polyhedra.     

On bifurcation of periodic solutions for functional differential equations of the neutral type with small delay

The class is singled out of systems described by ordinary differential equations unsolved relative to a derivative, in which a small delay leads to bifurcation of periodic solutions from the equilibrium state. The direct application of the classical results of M.A. Krasnosel’skii to these systems is made difficult in view of the complex character of the dependence on a bifurcation parameter, which is a small delay. The problem on bifurcation of periodic solutions for the stated systems is solved by methods of the theory of rotation of condensing vector fields.     

On one extremal problem of adaptive machine learning for detection of anomalies

An adaptive algorithm to solve a wide range of problems of unsupervised learning by constructing a sequence of interrelated extremal principles was proposed. The least squares method with a priori defined weights used as a starting point enabled determination of the “center” of learning sample. Next, a natural passage from the least squares method to more flexible extremal principle enabling adaptive determination of both the “center” and weights of the learning sample events was performed. Finally, a universal extremal principle enabling determination of the scaling coefficient of the membership function in addition to the “center” and weights was constructed.     

On convergence of kernel estimators of density with variable window width by dependent observations

In [1, 2] was studied a new type of nonparametric kernel estimators of probability density, whose window width varies depending on the sample, i.e., are data-based. These estimators were called adaptive. New estimators of density are superior in the rate of convergence to classical Rosenblatt-Parzen estimators. However, these valuable properties of estimators were obtained assuming that observations are independent. In this paper, we study properties of these adaptive estimators but assuming that the sample is realization of the stationary in the narrow sense random sequence. The simulation examples for the adaptive estimator constructed by dependent observations which is generated by autoregressive models are represented. The results of the investigation prove the advantage of the adaptive estimator over the classical Rosenblatt-Parzen estimator in the sense of the mean-square error. The rate of mean-square convergence of the limiting estimator (the so-called “ideal” estimator) to the true unknown density according to the dependent sample is found. The consistency of the adaptive estimator constructed by stationary dependent observations is proved.     

On instability of control systems for a certain class of objects and methods of its elimination

In practice of design of control systems, the cases occur when some of the roots of the transfer function of a controllable object are disposed on the imaginary axis of the complex plane. The optimal controller constructed for such objects, despite its realizability, will not afford the robustness properties in the system. The methods of removal of this phenomenon are given. The comparative estimate of the solution of this problem is provided both in the space of states and in the input-output relations (in the space of operators).