Perfect adaptation of perturbed nonlinear systems

A biological system is said to possess perfect adaptation ability if its biological function can restore precisely to the previous level after being affected by environmental changes. Such a biological system can be mathematically modelled as an input-output system whose output is asymptotically stable under all input disturbances concerned. Especially, in this paper, nonlinear systems whose dynamics are perturbed by disturbances through additive perturbation functions are used to investigate their perfect adaptation ability. First, using way of Lyapunov function, we give a sufficient condition for the single-output nonlinear systems with additive perturbation functions of constant disturbances to be perfectly adapting. Second, we survey the linear case of perfect adaptation systems. Furthermore, a biological perfect adaptation model and an example of nonlinear system are demonstrated to verify of the above results. Different with the existing mathematical studies on perfect adaptation, most of which are based on the viewpoint of control theory, we first interpret it by methods of nonlinear analysis.