Volterra Functional-Operator Equations in the Theory of Optimal Control of Distributed Systems

On preservation of global solvability of controlled second kind operator equation

“…We observe that the fail of the global solvability of an evolution controlled system associated with a differential or integro-differential equation is very likely, when the growth order of the right hand side in the corresponding equation with respect to the phase variable exceeds the linear growth, see demonstrative examples in [6], [8], [9,Introduction,Sect. 2].…”

“…In many situations one succeeds to prove that if, for instance, a system is globally solvable for some fixed control, then it keeps this property for all sufficiently small in a proper sense variations of this control; at the same time, for some admissible controls there can be no global solvability. Exactly this property accompanied by the uniqueness of the solution is called the stability of existence of global solutions or, more generally, the preservation of unique global solvability, see, for instance, the surveys in [26], [27], [28], [29].…”

“…Then corresponding abstract results were applied. More details on the history of developing the method of Volterra operator for obtaining SEGS conditions for controlled distributed systems can be found in the above cited surveys [26], [27], [28], [29]. In work [29], there was obtained an SEGS test for initial-boundary value problem related with a controlled semi-linear equation of a global electric circuit.…”