On Exact Global Controllability of a Semilinear EvolutionEquation with a Time-Varying Operator

For the Cauchy problem associated with a controlled semilinear evolution equation with bounded time-varying operator in a Hilbert space, we obtain sufficient conditions for the exact controllability to a given final state (and also to given intermediate states at intermediate time instants) on an arbitrarily (without additional conditions) fixed time interval. In fact, we generalize a similar result formerly obtained by the author for the case of a constant operator; here we use the Minty–Browder theorem and also the chaining technique of successive continuation of the solution of a control system to intermediate states. By way of example (which is of specific interest) we consider a semilinear equation describing the global electric circuit in the Earth atmosphere.