On exact observability of a nonlinear evolutionary equation with a bounded right-hand side operator on a small interval

For the Cauchy problem associated with a nonlinear ordinary differential equation in a Hilbert space $X$, we obtain sufficient conditions for exact observability on a small interval. By means of the condition of boundedness from below by a positive constant on the unit sphere with respect to a linear observer (observation operator) and with the help of Minty–Browder's theorem, the observability problem is reformulated as an operator (integral) equation with the right-hand side involving (in addition to the Volterra-type term, which is “local” in time) a nonlocal term as well. The unique solvability of the obtained operator equation (the equation of state reconstruction from observation) is proved with the help of the contraction map principle and the hypothesis about smallness of the observation interval. Moreover, we prove two theorems on global reconstruction of a state: 1) from observation on a small interval and under the condition of global solvability of some majorant integral equation in the space $\mathbb{R}$; 2) from a series of observations on small intervals in the presence of a priori information on the belonging of the state values to a bounded ball in $X$. As an example (of an independent interest), a semilinear equation of the global electric circuit in the Earth's atmosphere is considered.