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Does the following hold?

Let $x=0$ be an equilibrium point for the system $\dot x(t)=f(x(t))$ and suppose the existence and uniqueness conditions of solutions on $[t_0, +\infty)$ are satisfied. If $x=0$ is locally asymptotically stable in $\mathcal{D}\subset \mathbb{R}^n$, can one conclude that, for any given signal $\tilde x(t)$ satisfying $\lim_{t\rightarrow \infty}\tilde x(t)=0$, the trajectory of the differential system $\dot x=f(x)+\tilde x(t)$ with initial state in $\mathcal{D}$ satisfies $\lim_{t\rightarrow \infty} x(t)=0$?

If not, what further conditions should be imposed?

Any help would be greatly appreciated.

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No, it does not hold in general. The input $\tilde{x}$ may bring the state $x$ outside the domain of stability $\mathcal{D}$ before converging to zero. More generally, the system may be null-input stable but its gain may not be finite.

You are looking into input-to-state stability questions. The usual treatment involves the use of Lyapunov and Lyapunov-like functions. The literature is extensive because there are many many different cases that have to be treated separately.

The concept was studied extensively by Eduardo Sontag. You may want to look at his page http://www.math.rutgers.edu/~sontag/ for references. Lars Grüne has notes online http://numerik.mathematik.uni-bayreuth.de/~lgruene/iss-stuttgart04/

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