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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, Volume 28, Issue 4, Pages 531–548 (Mi vuu655)

MATHEMATICS

Majorant sign of the first order for totally global solvability of a controlled functional operator equation

A. V. Chernovab

a Nizhni Novgorod State University, pr. Gagarina, 23, Nizhni Novgorod, 603950, Russia
b Nizhni Novgorod State Technical University, ul. Minina, 24, Nizhni Novgorod, 603950, Russia

Abstract: We consider a nonlinear functional operator equation of the Hammerstein type which is a convenient form of representation for a wide class of controlled distributed parameter systems. For the equation under study we prove a solution uniqueness theorem and a majorant sign for the totally (with respect to a whole set of admissible controls) global solvability subject to Volterra property of the operator component and differentiability with respect to a state variable of the functional component in the right hand side. Moreover, we use hypotheses on the global solvability of the original equation for a fixed admissible control $u=v$ and on the global solvability for some majorant equation with the right hand side depending on maximal deviation of admissible controls from the control $v$. For example we consider the first boundary value problem associated with a parabolic equation of the second order with right hand side $f( t, x(t),u(t))$, $t=\{ t_0,\overline{t}\}\in\Pi=(0,T)\times Q$, $Q\subset\mathbb{R}^n$, where $x$ is a phase variable, $u$ is a control variable. Here, a solution to majorant equation can be represented as a solution to the zero initial-boundary value problem of the same type for analogous equation with the right hand side $bx^{q/2}+a_0x+Z$, where $Z(t)=\max\limits_{u\in\mathcal{V}(t)} |f(t,x[v](t),u)-f(t,x[v](t),v(t))|$, $\mathcal{V}(t)\subset\mathbb{R}^s$ is a set of admissible values for control at $t\in\Pi$, $q>2$, $s\in\mathbb{N}$; $a_0(.)$ and $b\geqslant0$ are parameters defined from $f^\prime_x$.

Keywords: functional operator equation of the Hammerstein type, totally global solvability, majorant equation, Volterra property.

DOI: https://doi.org/10.20537/vm180407

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Bibliographic databases:

UDC: 517.957, 517.988, 517.977.56
MSC: 47J05, 47J35, 47N10

Citation: A. V. Chernov, “Majorant sign of the first order for totally global solvability of a controlled functional operator equation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:4 (2018), 531–548

Citation in format AMSBIB
\Bibitem{Che18} \by A.~V.~Chernov \paper Majorant sign of the first order for totally global solvability of a controlled functional operator equation \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2018 \vol 28 \issue 4 \pages 531--548 \mathnet{http://mi.mathnet.ru/vuu655} \crossref{https://doi.org/10.20537/vm180407} \elib{http://elibrary.ru/item.asp?id=36873368}