User:Tibnor/ISS

In mathematics, the notion of input-to-state stability occurs in the study of dynamical systems with inputs. In simple therms, for all bounded inputs the state will be bounded, and if the bounds on the input decrease with time the bound on the state will also decrease.

Definition for continuously-time system
Consider an nonlinear dynamical system with input.


 * $$\dot{x} = f(t,x,u)), \;\;\;\; x(0) = x_0$$,

where $$x(t) \in \mathbb{R}^n$$ denotes the system state vector, $$u(t) \in \mathbb{R}^m$$ and $$f: [ 0, \infty ) \times \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$$ continuous on $$\mathcal{D}$$.  Suppose $$f$$ has an equilibrium $$x_e$$.