1,720,975 research outputs found
The method of unitary transformations in stability theory
No full textThe author suggests using unitary transformations of solutions to investigate the stability of solutions for linear and quasilinear differential systems with normal matrices of coefficients (AA^*=A^*A). This approach enables him to obtain differential inequalities for the square of the norm of a solution connected with the spectrum of the matrices of coefficients. Some examples are given to illustrate the method
An algorithm for constructing a quasi-regular asymptotic representation for the solution of singularly perturbed linear multi-point boundary value problems with fast and slow variables
No full textThe article is devoted to the system epsilondot x=A_{11}(t)x+A_{12}(t)z+f_1(t), dot z=A_{21}(t)x+A_{22}(t)z+f_2(t), where epsilon is a small positive parameter. The points 0=t_1<t_2<dots<t_n=1 are given. A solution x(t,epsilon),z(t,epsilon) of the mentioned system satisfying the multi-point boundary conditions sum^n_{j=1}F_jx(t_j,epsilon)=x^0, z(0,epsilon)=z^0 is to be found. The author gives sufficient conditions when a bounded solution of this problem as epsilonto0 exists. The form of the solution is obtained
Quasi-regular asymptotic behavior of the solution of a singularly perturbed Cauchy problem for linear systems of differential matrix equations
No full textThe author constructs an asymptotic expansion in powers of epsilon of the solution on a finite interval of t of the singularly perturbed matrix Cauchy problem epsilon dot{Z}=A(t)Z+ZB(t),quad Z(0,epsilon)=Z_0, where A(t) and B(t) are real square matrices depending smoothly on t. It is assumed that the eigenvalues of the matrices are simple and have non-positive real parts. A norm estimate of the error term is given. An expansion is also given for the solution of the non-homogeneous Cauchy problem
A spectral method for studying the stability of some classes of nonautonomous differential equations
No full textThe author proposes a method to investigate (without using Lyapunov functions) the stability of the trivial solution of differential systems of three types: (1) dot x=A(t,epsilon)x+f(x,t), x(0,epsilon)=x^0, where A(t,epsilon)=sum^infty_{k=0}A_k(t)epsilon^k, f(0,t)equiv 0, |A(t,epsilon)|leq C, tgeq 0, |epsilon|leqepsilon_0<1; (2) epsilondot x=A(t,epsilon)x+epsilon b(x,t), x(0,epsilon)=x^0, where the series A(t,epsilon)=sum^infty_{k=0}A_k(t)epsilon^k is absolutely and uniformly convergent for tgeq 0, |epsilon|leqepsilon_0; (3) dot x=A(t)x+f(x,t), x(t_0)=x^0, where A(t)=t^msum^infty_{k=0}A_k(t)t^{-k}, mgeq 1, f(0,t)equiv 0, tgeq t_0geq 1. (For all systems the A_k are sufficiently smooth T-periodic matrix functions.) par Conditions for stability, asymptotic stability and instability are obtained. Some examples are given to illustrate the proposed methods
Construction of exact solutions to singular perturbation problems for linear ordinary differential equations with power boundary layer
No full text[No abstract available
On a method for studying the norm and the stability of solutions
No full textWe present a new method (the method of unitary transformations), which differs from the existing ones, for studying the stability and the norm of solutions of regular and singularly perturbed initial-value problems for nonautonomous linear and quasilinear systems of ODE with normal and "almost normal" matrices. Our results generalize similar theorems for the corresponding systems with constant matrices. This method allows one to avoid rather cumbersome traditional analysis, including the Lyapunov function method. For special classes of singularly perturbed problems, the method provides estimates for the norms of solutions in the presence of exponential or power boundary layers; these observations enrich the collection of known results in this field. © Nauka/Interperiodica 2007
On a method for studying the norm and the stability of solutions
No full textWe present a new method (the method of unitary transformations), which differs from the existing ones, for studying the stability and the norm of solutions of regular and singularly perturbed initial-value problems for nonautonomous linear and quasilinear systems of ODE with normal and "almost normal" matrices. Our results generalize similar theorems for the corresponding systems with constant matrices. This method allows one to avoid rather cumbersome traditional analysis, including the Lyapunov function method. For special classes of singularly perturbed problems, the method provides estimates for the norms of solutions in the presence of exponential or power boundary layers; these observations enrich the collection of known results in this field. © Nauka/Interperiodica 2007
On the structure of the solution of singularly perturbed initial boundary value problems with an unbounded spectrum of the limit operator
No full textSingularly perturbed initial boundary value problems are studied for some classes of linear systems of ordinary differential equations on the semiaxis with an unbounded spectrum of the limit operator. We give a new version of the proof of the existence of a unique and bounded (as ε → +0) solution for which with the help of the splitting method we construct a uniform asymptotic expansion on the entire semiaxis and describe all singularities (reflecting the structure of the corresponding boundary layers) in closed analytic form, including the critical case in which the points of the spectrum of the limit operator can touch the imaginary axis; this supplements previous results. ©1999 Kluwer Academic/Plenum Publishers
On the structure of the solution of singularly perturbed initial-boundary value problems with an unbounded spectrum of the limit operator
No full textSummary (translated from the Russian): "We consider singularly perturbed initial-boundary value problems for some classes of linear systems of ordinary differential equations on the half-line with an unbounded spectrum of the limit operator. We present a new version of the proof of the existence of a unique and bounded solution as tto +0 for which, using the splitting method, we construct an asymptotic expansion, uniform on the entire half-line, and describe all the singularities (reflecting the structure of the corresponding boundary layers) in closed analytic form, including the critical case when the spectrum points of the limit operator may be tangent to the imaginary axis, which supplements previous results.
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