Maximal linear transformation groups preserving asymptotic properties of linear differential systems: I

We consider the problem of describing sets of linear piecewise differentiable transformations that preserve some asymptotic property of linear differential systems. We present definitions needed for solving this problem, obtain preliminary results, and describe the set of linear transformations preserving the property of boundedness of the coefficients of linear differential systems on the time half-line.     

A dynamical systems result on asymptotic integration of linear differential systems

We are interested in the asymptotic integration of linear differential systems of the form x′=[Λ(t)+R(t)]x, where Λ is diagonal and R∈L^p[t₀,∞) for p∈[1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ω_Λ. Our results include examples that are not covered by the Hartman-Wintner theorem.     

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.     

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.      

Structure of stability and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: II

We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying the matrix of the system. The asymptotic stability set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are asymptotically stable. We prove that a set on the real axis is the asymptotic stability set of such a family if and only if it is an F _σδ -set lying entirely on an open ray with origin at zero. In addition, for any set of this kind, the coefficient matrix of a family whose asymptotic stability set coincides with this set can be chosen to be infinitely differentiable and uniformly bounded on the time half-line.     

On asymptotic equivalence of perturbed linear systems of differential and difference equations

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off-diagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman-Wintner. Some analogous results for systems of linear difference equations are also given.     

On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients

A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients have the formt ^-α a(t), α > 0 wherea(t) is a trigonometric polynomial with an arbitrary set of frequencies. The asymptotic behavior of the solutions of this system ast → ∞ is studied. We construct an invertible (for sufficiently larget) change of variables that takes the original system to a system not containing oscillatory coefficients in its principal part. The study of the asymptotic behavior of the solutions of the transformed system is a simpler problem. As an example, the following equation is considered:

Methods for studying the asymptotic properties of systems with two linear delays

We study methods for the instability analysis of neutral-type systems with two delays linearly depending on the argument. In the case of instability of the neutral part of a system with constant coefficients, we suggest a stabilization algorithm. In addition, by using the Laplace transform, we obtain sufficient conditions for the instability of the solution of a given system.     

Boundedness of solutions and asymptotic properties of some systems of differential equations

We establish conditions under which solutions of some systems of differential equations are bounded and study their asymptotic properties.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 944–946, July, 1994.     

On preserving dissipativity properties of linear complementarity dynamical systems with the \theta -method

In this work we study the following problem: given a numerical method (an extended $\theta $ -method named the $(\theta , \gamma )$ -method), find the class of dissipative linear complementarity systems such that their discrete-time counterpart is still dissipative, with the same storage (energy) function, supply rate (reciprocal variables), and dissipation function. Systems with continuous solutions, and with state jumps are studied. The notion of numerical dissipation is given a rigorous meaning.