On convexity local conditions for attainable tubes of controlled distributed systems

For a nonlinear functional operator equation being a form of describing for a wide class of controlled initial boundary-value problems, we introduce the concept of abstract attainable set as an analog of the attainable tube concept. We obtain local sufficient conditions for the convexity of such a set. Using the general results is illustrated by example of the mixed boundary-value problem associated with a semilinear hyperbolic equation of the second order and a rather general form.