The standard theory of Lyapunov stability for the system of ordinary differential equations x′(t)=f(t,x(t)), f∈C([0,∞)×RN;RN), says that if there exists a function V(t,x) such that (1) W1(|x|)≤V(t,x)≤W2(|x|) and (2) (d/dt)V(t,x(t))≤−W(|x|) for a solution x(t), then x(t) tends to 0 as t→∞, where Wi,W denote "wedge'' functions. We generalize the above result, replacing −W(|x|) in (2) by φ(t)−W(Vˆ)k(t) with some specific functions φ(t), k(t) and Vˆ. Many previous theorems are reduced to special cases of our general results.
Remarks on Lyapunov stability
PUCCI, Patrizia;
1995
Abstract
The standard theory of Lyapunov stability for the system of ordinary differential equations x′(t)=f(t,x(t)), f∈C([0,∞)×RN;RN), says that if there exists a function V(t,x) such that (1) W1(|x|)≤V(t,x)≤W2(|x|) and (2) (d/dt)V(t,x(t))≤−W(|x|) for a solution x(t), then x(t) tends to 0 as t→∞, where Wi,W denote "wedge'' functions. We generalize the above result, replacing −W(|x|) in (2) by φ(t)−W(Vˆ)k(t) with some specific functions φ(t), k(t) and Vˆ. Many previous theorems are reduced to special cases of our general results.File in questo prodotto:
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