Abstract—We find an estimate on the norm of the operator that describes the dynamics of a linear two-layered one-dimensional unilaterally bounded heat transfer object with nonperfect heat contact between the layers. Based on this estimate, we find a sufficient causality and stability condition for the closed system arising when the object is closed with feedback.

Modern problems of optimization, estimation, signal and image processing, pattern recognition, etc., deal with huge-dimensional data; this necessitates elaboration of efficient methods of processing such data. The idea of building low-dimensional approximations to huge data arrays is in the heart of the modern data analysis. One of the most appealing methods of compact data representation is the statistical method referred to as the principal component analysis; however, it is sensitive to uncertainties in the available data and to the presence of outliers. In this paper, robust versions of the principle component analysis approach are proposed along with numerical methods for their implementation.

The stability of nonlinear cascade systems with a time delay is studied. Conditions of the global asymptotic stability in terms of linear matrix inequalities for a finite set of matrices are obtained. The problem of stabilization of the controlled delay system is considered, which is solved based on the stability conditions. The proposed approach to the analysis of qualitative properties and to the solution of stabilization problems is based on the results concerning the asymptotic stability of the delay linear systems, the decomposition of the original system, and the representation of the delay nonlinear system by a Takagi–Sugeno system. Examples illustrating the simplification of the system analysis by reducing its size and decreasing the number of linear matrix inequalities are discussed.

For multidimensional discrete mappings of the form s_{k+1} = g(s_k, p) we develop the method of symbolic CTQ-analysis and study the main properties of the T-alphabet. We formulate basic principles for the analytic study of discrete mappings in terms of the proposed formalism.