Vol 25, No 130 (2020)
Articles
Statistical filtering algorithms for systems with random structure
Abstract
New algorithms for solving the optimal filtering problem for continuous-time systems with a random structure are proposed. This problem is to estimate the current system state vector from observations. The mathematical model of the dynamic system includes nonlinear stochastic differential equations, the right side of which defines the system structure (regime mode). The right side of these stochastic differential equations may be changed at random time moments. The structure switching process is the Markov or conditional Markov random process with a finite set of states (structure numbers). The state vector of such system consists of two components: the real vector (continuous part) and the integer structure number (discrete part). The switch condition for the structure number may be different: the achievement of a given surface by the continuous part of the state vector or the distribution of a random time period between structure switchings. Each ordered pair of structure numbers can correspond to its own switch law. Algorithms for the estimation of the current state vector for systems with a random structure are particle filters, they are based on the statistical modeling method (Monte Carlo method). This work continues the authors’ research in the field of statistical methods and algorithms for the continuous-time stochastic systems analysis and filtering.
Russian Universities Reports. Mathematics. 2020;25(130):109-122
109-122
On the spectral properties and positivity of solutions of a periodic boundary value problem for a second-order functional differential equation
Abstract
For a functional-differential operator Lu = (1/ρ) -(pu')' + 0 l u(s) d s r(x, s) with symmetry, the completeness and orthogonality of the eigenfunctions is shown. The positivity conditions of the Green function of the periodic boundary value problem are obtained.
Russian Universities Reports. Mathematics. 2020;25(130):123-130
123-130
The problem of boundary control of string vibrations by displacement of the left end when the right end is fixed with the given values of the deflection function at intermediate times
Abstract
We consider the boundary control problem for the homogeneous string vibration equation with given the classical boundary (initial and final) conditions and with given values of the deflection function at intermediate times. The control is performed by displacement of the left end of the string when the right end is fixed. The problem is reduced to the control problem with zero boundary conditions. We propose the constructive method for constructing the boundary control of the process of string vibrations with given values of the deflection function at intermediate times.We present the results of numerical experiments and the corresponding graphs confirm the validity of the results.
Russian Universities Reports. Mathematics. 2020;25(130):131-146
131-146
On the possibility of obtaining the optimal order of accuracy when restoring the impact by the dynamic method
Abstract
Osipov and A. V. Kryazhimsky proposed a method of dynamic regulation to restore an unknown effect in a controlled model. In the framework of this approach, in the present work we study the property of another method based on the use of the implicit Euler method for the problem of numerical differentiation. The choice of the parameters of the method is indicated, which makes it possible to increase its efficiency, reduce the noise level of the approximate solution, and obtain the optimal order of accuracy in the metric L(T) ; equal to 1 2 .
Russian Universities Reports. Mathematics. 2020;25(130):147-155
147-155
Ob ustoychivom priblizhennom reshenii odnoy nekorrektno postavlennoy kraevoy zadachi dlya metagarmonicheskogo uravneniya
Abstract
In this paper, we consider a mixed problem for a metaharmonic equation in a region in a rectangular cylinder. On the side faces cylinder region is set to homogeneous conditions of the first kind. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i. e. the function and its normal derivative are set. The other boundary of the cylindrical region, which is flat, is free. This problem is illposed, and to construct its approximate solution in the case of Cauchy data known with some error, it is necessary to use regularizing algorithms. In this paper, the problem is reduced to the Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained. A stable solution of the integral equation is obtained by the method of Tikhonov regularization. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem as a whole is constructed. The convergence theorem of the approximate solution of the problem to the exact one is given when the error in the Cauchy data tends to zero and when the regularization parameter is agreed with the error in the data. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.
Russian Universities Reports. Mathematics. 2020;25(130):156-164
156-164
The conditions of minimum for a smooth function on the boundary of a quasidifferntiable set
Abstract
In this paper, we consider problems of mathematical programming with nonsmooth constraints of equality type given by quasidifferentiable functions. By using the technique of upper convex approximations, developed by B. N. Pshenichy, necessary conditions of extremum for such problems are established. The Lagrange multipliers signs are specified by virtue of the fact that one can construct whole familers of upper convex approximations for quasidifferentiable function and thus the minimum points in such extremal problems are characterized more precisely. Also the simplest problem of calculus of variations with free right hand side is considered, where the left end of the trajectory starts on the boundary of the convex set. The transversality condition at the left end of the trajectory is improved provided sertain sufficient conditons hold.
Russian Universities Reports. Mathematics. 2020;25(130):165-182
165-182
Properties of the algebra Psd related to integrable hierarchies
Abstract
In this paper we discuss and prove various properties of the algebra of pseudo differential operators related to integrable hierarchies in this algebra, in particular the KP hierarchy and its strict version. Some explain the form of the equations involved or give insight in why certain equations in these systems are combined, others lead to additional properties of these systems like a characterization of the eigenfunctions of the linearizations of the mentioned hierarchies, the description of elementary Darboux transformations of both hierarchies and the search for expressions in Fredholm determinants for the constructed eigenfunctions and their duals.
Russian Universities Reports. Mathematics. 2020;25(130):183-195
183-195
Relaxation of the game problem of guidance connected with alternative in guidance-evasion differential game
Abstract
Differential game (DG) of guidance-evasion for a finite time interval is considered; as parameters, the target set (TS) and the set defining phase constraints (PC) are used. Player I interested in realization of guidance with TS under validity PC uses set-valued quasistrategies (nonanticipating strategies) and Player II having opposite target uses strategies with nonanticipating choice of correction instants and finite numbers of such instants. On informative level, the setting corresponds to alternative theorem of N. N. Krasovskii and A. I. Subbotin. For position not belonging to solvability set of Player I, determination of the least size of neighborhoods for set-parameters under that Player I guarantees guidance (under weakened conditions) is interested. In article, this scheme is supplemented by priority elements in questions of TS attainment and PC validity; this is realized by special parameter defining relation for sizes of corresponding neighborhoods. Under these conditions, a function of the least size of TS neighborhood is defined by procedure used program iteration method for two variants. The above-mentioned function is fixed point for one of two used “program” operators. Special type of the quality functional for which values of the above-mentioned function coincide with values of the minimax-maximin games is established.
Russian Universities Reports. Mathematics. 2020;25(130):196-244
196-244