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

We consider families of linear differential systems continuously depending on a real parameter. The stability (respectively, asymptotic stability) set of such a family is defined as the set of all values of the parameter for which the corresponding systems in the family are stable (respectively, asymptotically stable). We show that a set on the real axis is the stability (respectively, asymptotic stability) set of some family of this kind if and only if it is an F _σ -set (respectively, an F _σδ -set). For families in which the parameter occurs only as a factor multiplying the matrix of the system, their stability sets are exactly F _σ -sets containing zero on the real line. The asymptotic stability sets of such families will be described in the second part of the present paper.     

Closed irregularity sets of linear differential systems with a parameter multiplying the derivative

For any closed subset M of the real line that does not contain zero, we construct a linear differential system with bounded piecewise continuous coefficient matrix A(·) such that the corresponding system with coefficient matrix μA(·) linearly depending on a real parameter μ is Lyapunov irregular for all μ in M and Lyapunov regular for all other parameter values.     

On the improperness sets of families of linear differential systems

We consider families of linear differential systems continuously depending on a real parameter with continuous (or piecewise continuous) coefficients on the half-line. The improperness set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are Lyapunov improper. We show that a subset of the real axis is the improperness set of some family if and only if it is a G _δσ -set. The result remains valid for families in which the matrices of the systems are bounded on the half-line. Almost the same result holds for families in which the parameter occurs only as a factor multiplying the system matrix: their improperness sets are the G _δσ -sets not containing zero. For families of the last kind with bounded coefficient matrix, we show that their improperness set is an arbitrary open subset of the real line.     

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.      

Steplike contrast structures for a second-order ordinary differential equation with a small parameter multiplying the highest derivative

A boundary value problem is considered for a second-order nonlinear ordinary differential equation with a small parameter multiplying the highest derivative. The limit equation has three solutions, of which two are stable and are separated by the third unstable one. For the original problem, an asymptotic expansion of a solution is studied that undergoes a jump from one stable root of the limit equation to the other in the neighborhood of a certain point. A uniform asymptotic approximation of this solution is constructed up to an arbitrary power of the small parameter.     

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.     

Absolute asymptotic stability of solutions of linear parabolic differential equations with delay

We establish necessary and sufficient conditions for the absolute asymptotic stability of solutions of linear parabolic differential equations with delay. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1714–1721, December, 2007.     

Remarks concerning the asymptotic stability and stabilization of linear delay differential equations

This note concerns the asymptotic stability for all values of the delays of controlled and uncontrolled linear delay differential equations. In the case of uncontrolled systems it is shown by example that structural considerations must be accounted for. In the controlled case, although arbitrary pole placement may not be possible, there is an algorithm which sometimes reduces the spectrum of the feedback system to one which is finite and is contained in the left half plane.     

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.     

Lower bounds for the norm of solutions of linear differential systems with a linear parameter

For a special class of two-dimensional linear differential systems $\dot x = \left( {A\left( t \right) + \mu {\rm B}\left( t \right)} \right)x$ including Lyapunov irregular almost periodic systems constructed by V.M. Millionshchikov, we prove the nonexistence of upper bounds for the norms of solutions uniform with respect to t ≥ 0 and µ ∈ ?.     

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:

On sets of linear differential systems to which perturbed linear systems cannot be reduced

The paper [2] defines the noncoinciding irreducibility sets N ₂(a, σ) and N ₃(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N ₂(a, σ)] the condition