A Stabilization Indices Problem

Abstract

We construct families of homogeneous linear differential equations (with coefficients continuously depending on a parameter) possessing certain topologically generic (in the sense of R. Baire) properties, such as the absence of various types of semicontinuity of their various conditional stability characteristics.In [1], V. M. Millionshchikov showed that for a systemẋ = A(μ, t)x with coefficients continuously depending on the parameter μ in a complete metric space, the upper semicontinuity of its Lyapunov exponents as functions of μ is a generic property.The term generic is understood here in the sense of R. Baire [2] (the French mathematician who introduced and studied this notion): a property of a point in a topological space is called Baire generic if the set of points having this property contains a dense G δ -set (an intersection of countably many open sets [3, p. 159]).In this paper, we construct families of linear systems (with coefficients continuously depending on a parameter) having a somewhat opposite generic property: various asymptotic characteristics of their solutions are lacking in one or another type of semicontinuity. Moreover, we give answers to some questions raised in [4].Given n ∈ N, denote by S n the space of linear systemṡwith continuous (not necessarily bounded) coefficients. The space S n is endowed with the uniform metric ρ(A, B) = min sup t∈R + A(t) − B(t) , 1 ,where Y = sup |x|=1