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Home » Issues » Archive of Issues » 2018 - 4 issue » pp. 531-548

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Russia Nizhni Novgorod
Year
2018
Volume
28
Issue
4
Pages
531-548
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Section Mathematics
Title Majorant sign of the first order for totally global solvability of a controlled functional operator equation
Author(-s) Chernov A.V.ab
Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb
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 $f\bigl( t, x(t),u(t)\bigr)$, $t=\{ t_0,\overline{t}\}\in\Pi=(0,T)\times Q$, $Q\subset\mathbb{R}^n$, 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
UDC 517.957, 517.988, 517.977.56
MSC 47J05, 47J35, 47N10
DOI 10.20537/vm180407
Received 23 May 2018
Language Russian
Citation Chernov A.V. Majorant sign of the first order for totally global solvability of a controlled functional operator equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 531-548.
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