Open AccessSolvability and construction of solution to the de la Vallee – Poussin problem for the second-order matrix Lyapunov equation with a parameter
Author(s) -
A. I. Kashpar,
V. N. Laptinskiy
Publication year - 2019
Publication title -
vescì nacyânalʹnaj akadèmìì navuk belarusì. seryâ fìzìka-matèmatyčnyh navuk
Language(s) - English
Resource type - Journals
eISSN - 2524-2415
pISSN - 1561-2430
DOI - 10.29235/1561-2430-2019-55-1-50-61
Subject(s) - mathematics , constructive , lyapunov function , rate of convergence , mathematical analysis , matrix (chemical analysis) , convergence (economics) , boundary value problem , nonlinear system , channel (broadcasting) , physics , materials science , process (computing) , quantum mechanics , economic growth , computer science , electrical engineering , economics , composite material , engineering , operating system
The paper considers the issues of constructive analysis of the de la Vallee – Poussin boundary-value problem for the second-order linear matrix differential Lyapunov equation with a parameter and variable coefficients. The initial problem is reduced to an equivalent integral problem, and to study its solvability a modification of the generalized contraction mapping principle is used. A connection between the approach used and the Green’s function method is established. The coefficient sufficient conditions for the unique solvability of this problem are obtained. Using the Lyapunov – Poincaré small parameter method, an algorithm for constructing a solution has been developed. The convergence and the rate of convergence of this algorithm have been investigated, and a constructive estimation of the region of solution localization is given. To illustrate the application of the results obtained, the linear problem of steady heat conduction for a cylindrical wall, as well as a two-dimensional matrix model problem is considered. With the help of the developed general algorithm, analytical approximate solutions of these problems have been constructed and on the basis of their exact solutions a comparative numerical analysis has been carried out.