On exact global controllability of nonlinear evolution equation in a Hilbert space under bounded control

For a Cauchy problem associated with a nonlinear ordinary differential equation in a Hilbert space $X$, we obtain sufficient conditions for exact controllability to a given final state (as well as to given intermediate states at intermediate times) over arbitrarily fixed (without additional conditions) time interval under a constraint on the control norm value. This is a generalization of a similar result previously obtained by the author for the case of an operator differential equation with a stationary linear operator and linearly incoming control without a constraint on the norm. As before, the Minty–Browder theorem is used, as well as the chain technology for sequentially continuing the solution of the control system to intermediate states. As an example (of independent interest), a strongly nonlinear pseudoparabolic partial differential equation describing the evolution of an electric field in a semiconductor is considered.