On the structure of a solution set of controlled initial-boundary value problems

For a controlled nonlinear functional-operator equation of the Hammerstein type describing a wide class of controlled initial-boundary value problems, we obtain simple sufficient conditions for the convexity, pointwise boundedness and precompactness of the set of solutions (the reachability tube) in the Lebesgue space. As concerns boundedness and precompactness, we mean certain conditions of the majorant type without Volterra type requirements which give also the total (with respect to the whole set of admissible controls) preservation of solvability of mentioned equation. In the capaсity of examples of reduction of a controlled initial-boundary (boundary) value problem to the equation under investigation and verification the proposed hypotheses for this equation, we consider the first initial-boundary value problem associated with a semilinear parabolic equation of the second order in a rather general form, and also the Dirichlet problem associated with a semilinear elliptic equation of the second order.