every practical system is a nonlinear system . so we have to understand the control of this systems . this systems can be approximated to a linear system and controlling of linear system we know very well, we can do pole placement according to our desired response criteria.
so nonlinear system can be approximated to linear system at the equilibrium point. equilibrium points are the points where all states are constant or not changing, find that point and linearize the system for that point and do the rest of the analysis same as the linear system.
stability can be BIBO stability or stability in terms of the equilibrium point, in BIBO stability for a bounded input there should be a bounded output but for stability in sense of equilibrium point there should be convergence of system should be present when we disturb the system means system should come back to the equilibrium point if we disturb the system. this is useful when we don’t know the input for the system then we can use this stability condition.
solution of the any LTI system we know that is the addition of the zero input response and zero state response. for the stability of the system this both response sound be bounded. if we solve this one then we can see the exponential term in that so if it decays then it will decay exponentially. that is called asymptotically stable, hence the exponential stability and asymptotically stability are same. either it will decay asymptotically or it will be unstable.