On the exponential stability of a state-dependent delay equation. (English) Zbl 0966.34068
The exponential stability of the trivial solution to the state-dependent delay differential equation
\[
\dot x (t) =a(t)x(t-\tau (t,x(t)))
\]
is investigated. It is shown that, under some conditions, this state-dependent equation is exponentially stable, if the trivial solution to
\[
\dot y (t)=a(t)y(t-\tau (t,0))
\]
is exponentially stable. Assuming the existence of bounded partial derivatives of the delay function, the reverse statement is proved.
Reviewer: Vasile Dragan (Bucuresti)
MSC:
34K20 | Stability theory of functional-differential equations |
34D20 | Stability of solutions to ordinary differential equations |
34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |