On the convexity of global solvability sets for controlled initial-boundary value problems

We consider a controlled functional-operator equation that is a convenient form for describing of controlled initial-boundary value problems. For this equation, considered in a Banach ideal space, we define the set Omega of global solvability as the set of all admissible controls for which the equation has a global solution. We show that Omega is convex under the conditions imposed on the right-hand side of the equation. For each control in a given segment in Omega, we obtain a two-sided pointwise estimate for the corresponding solution under the abovementioned assumptions. We prove the theorem on the convex continuous dependence of the solution on the parameter specifying the displacement along the segment.