Preservation of the Solvability of a Semilinear Global Electric Circuit Equation

On preservation of global solvability of controlled second kind operator equation

“…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].…”

“…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. Here the same idea on reducing to a Volterra functional-operator equation was employed; the equation was of Hammerstein type.…”

“…The equation was considered in a space (︀ [0, ]; )︀ with some Banach space of functions defined on a domain Ω ⊂ R . 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 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.…”