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Stability of linear systems

Profile image of Sándor KovácsSándor Kovács

2018

Abstract

This paper examines the relation of the exponential dichotomy and the stability concepts for systems of linear differential equations. We are going to show some relationship between the studied concepts, more precisely we are presenting how the stability of a linear non-autonomous system is investigated with the help of the exponential dichotomy. Furthermore we are going to show how the stable and unstable subspace of an exponentially dichotomic system can be specified using the definition of the exponential dichotomy.

Figures (3)
With the above two inequalities we can show the boundedness of the solution y. Because for each t € R the estimation Finally one can prove the uniqueness of y by the superposition principle from the uniqueness in (ii). Conversely if (iii) holds, we can get with f =0 (or by the superposition principle) that statement (iii) also holds.
With the above two inequalities we can show the boundedness of the solution y. Because for each t € R the estimation Finally one can prove the uniqueness of y by the superposition principle from the uniqueness in (ii). Conversely if (iii) holds, we can get with f =0 (or by the superposition principle) that statement (iii) also holds.
Figure 1. The phase portrait of system (3.9). It can be seen that Example 3.5. Let us consider the following autonomous system:
Figure 1. The phase portrait of system (3.9). It can be seen that Example 3.5. Let us consider the following autonomous system:
In Figure 3.5 the norm ||-||2 of the solution y is shown on the interval t € [0, 6]. The asymptotic properties (3.6) hold for the solution (3.10), but as also it is outlined in Figure 2, the solution is not monotone decreasing. Based on Proposition 3.3 we have made some remarks on the stable and unstable subspaces and the topologically equivalence of autonomous linear sys- tems.
In Figure 3.5 the norm ||-||2 of the solution y is shown on the interval t € [0, 6]. The asymptotic properties (3.6) hold for the solution (3.10), but as also it is outlined in Figure 2, the solution is not monotone decreasing. Based on Proposition 3.3 we have made some remarks on the stable and unstable subspaces and the topologically equivalence of autonomous linear sys- tems.

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References (12)

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