Vol 25, No 130 (2020)

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.

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.

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.

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.