Abstract
This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.
Highlights
It is well known that Lyapunov [1] laid the foundation of stability theory
We establish a relationship between the ith curvature (i = 1, 2, · · · ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix
In [2], the authors calculated the curvature and torsion of the state trajectories r(t) of the two- and three-dimensional linear timeinvariant systems ṙ(t) = Ar(t), which are related to the system stability
Summary
It is well known that Lyapunov [1] laid the foundation of stability theory. Linear systems are the most basic and widely used research objects, which have been developed for a long period. We will describe the stability of the zero solutions of linear time-invariant systems in arbitrary dimension by using the torsion, namely, the second curvature. Suppose that ṙ (t) = Ar (t) is a linear time-invariant system, where A is an n × n invertible real matrix, and r (t) ∈ Rn. Denote by τ (t) the torsion of trajectory of a solution r (t).
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