The Total Preservation of Unique Global Solvability of the First Kind Operator Equation With Additional Controlled Nonlinearity

On preservation of global solvability of controlled second kind operator equation

“…At the same time, the technique of [29] can be applied to semi-linear equations only. In [30], there was proved a condition for total (over all admissible controls) preservation of global solvability of evolution operator equation of first kind of general form with a controlled additional nonlinearity in a Banach space. The technique of [30] can be applied also to essentially nonlinear partial differential equations of evolution type.…”

“…In [30], there was proved a condition for total (over all admissible controls) preservation of global solvability of evolution operator equation of first kind of general form with a controlled additional nonlinearity in a Banach space. The technique of [30] can be applied also to essentially nonlinear partial differential equations of evolution type. The matter of the condition in [30] is the total preservation of global solvability (TPGS) of the mentioned equation was guaranteed by assuming the global solvability for some majorizing uniformly in all controls integral equation for an unknown function depending on the time variable only.…”

“…The technique of [30] can be applied also to essentially nonlinear partial differential equations of evolution type. The matter of the condition in [30] is the total preservation of global solvability (TPGS) of the mentioned equation was guaranteed by assuming the global solvability for some majorizing uniformly in all controls integral equation for an unknown function depending on the time variable only. At that, the majorizing equation was constructed on the base of postulated estimates for the controlled additional nonlinearity.…”

“…The present paper is a result of an essential revisiting of works [29], [30]. First, the results of [29], [30] can be reformulated as conditions for SEGS and TPGS, respectively, for an evolution operator equation of second kind and in fact, such approach is more natural. Second, the space ([0; ]; ) serving in the cited works as the space, in which the solution is sought, can be poorly adapted for satisfying made assumptions.…”

“…Second, the space ([0; ]; ) serving in the cited works as the space, in which the solution is sought, can be poorly adapted for satisfying made assumptions. Third, the results of [29], [30] allow one to estimate the solution only in the norm of the space ([0; ]; ). At the same time, in applications, one can need to estimate in norm of the space [0; ] (say, Sobolev space) and to seek the solution in this space.…”