In connection with state determination, the observability problem of nonlinear dynamical systems has been studied [l-5]. The observability problem is to find conditions under which a knowledge of the input-output data uniquely determines the state of the system. Griffith and Kumar  treated this problem and reported some necessary conditions and a sufficient condition of observability of nonlinear dynamical systems. This paper treats autonomous nonlinear dynamical systems. In particular, for the construction of optimal control, it is assumed that the optimal control is a function of state. In this case, time-invariant nonlinear dynamical systems having inputs turn into time-invariant nonlinear dynamical systems without inputs, i.e., autonomous nonlinear dynamical systems. This paper classifies these autonomous nonlinear systems into three types of systems, that is, Ck type, analytic type, and polynomial type, and gives a necessary and sufficient condition of observability for each type of system.
Inouye, Y. (1977). On the observability of autonomous nonlinear systems. Journal of Mathematical Analysis and Applications, 60(1), 236–247. https://doi.org/10.1016/0022-247X(77)90062-2