On uniform asymptotic <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>h</a:mi> </a:math> -stability for linear non autonomous systems.

A classical result in the theory of stability of nonautonomous linear systems is that the uniform exponential decay of its corresponding transition matrix is equivalent to the uniform asymptotic stability of the trivial solution. The aim of this article is to extend this equivalence for non-exponential decays <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:math> . In this article we prove that, in some suitable cases, the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:math> allows the construction of a topological abelian group that makes possible to formulate a more general definition of uniform stability which is equivalent with a decay dominated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:math> . Moreover, we can use this group to establish the necessary elements to develop a theory of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -stable systems. As a first step in this direction, we provide integral conditions on the solutions of a uniformly asymptotically <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -stable system.