Output tracking and steady state for nonlinear systems

The problem driving this work is exact output tracking using stable inversion for nonlinear systems (models of aircraft). The problem of stable inversion evolves to a time varying nonlinear system with inputs and the search for a unique bounded continuous solution on (-∞,∞) in response to a bounded input on (-∞,∞). We want to show that all bounded continuous solutions on 0⩽t<∞ converge to this unique one as t→∞ under the appropriate assumptions. The bounded and continuous solution on (-∞,∞) can be thought of as a nonlinear steady state. This exactly parallels the idea of steady state for a time invariant linear system whose homogenous part has no eigenvalues on the imaginary axis. In response to a bounded input on (-∞,∞) is the unique bounded and continuous solution on (-∞,∞). All bounded and continuous solutions on 0⩽t<∞ must converge to this unique one, which is called the steady state. Hence, under appropriate assumptions, steady state solutions exist for nonlinear systems even though the principle of superposition does not hold