Operator Equations of the Second Kind: Theorems on the Existence and Uniqueness of the Solution and on the Preservation of Solvability
This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator acting on an arbitrary Banach space , new theorems on the existence and uniqueness of a fixed po...
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| Vydané v: | Differential equations Ročník 58; číslo 5; s. 649 - 661 |
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01.05.2022
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| Abstract | This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator
acting on an arbitrary Banach space
, new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used:
, where
is, generally speaking, another Banach space semi-ordered by the cone
. These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm
is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov (
,
). The role of an analog of the Lipschitz constant on a given bounded set
is played by a bounded linear operator
, depending on this set, with spectral radius
. In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation
,
, where
is the control parameter from, generally speaking, an arbitrary set
. The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation. |
|---|---|
| AbstractList | This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator
acting on an arbitrary Banach space
, new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used:
, where
is, generally speaking, another Banach space semi-ordered by the cone
. These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm
is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov (
,
). The role of an analog of the Lipschitz constant on a given bounded set
is played by a bounded linear operator
, depending on this set, with spectral radius
. In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation
,
, where
is the control parameter from, generally speaking, an arbitrary set
. The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation. This paper continues the author's research on the problem of preserving the solvability ofcontrolled operator equations. As a preliminary result (which is of independent interest) for ageneral operator [Formula omitted] acting on an arbitrary Banach space [Formula omitted], new theorems on the existence and uniqueness of afixed point are obtained. Here the well-known concept of the cone norm is used: [Formula omitted],where [Formula omitted] is, generally speaking, another Banach spacesemi-ordered by the cone [Formula omitted].These theorems are based on the assumption that the operator analog of the local Lipschitzcondition with respect to the cone norm [Formula omitted] is satisfied andgeneralize the result by A.V. Kalinin and S.F. Morozov ( [Formula omitted], [Formula omitted]).The role of an analog of the Lipschitz constant on a given bounded set [Formula omitted] is played by a bounded linear operator [Formula omitted], depending on this set,with spectral radius [Formula omitted]. Inaddition, M.A. Krasnosel'skii's lemmas on the equivalent norm are used. Based on the statementsobtained, we prove theorems on local and total (over the set of admissible controls) preservation ofthe solvability of the controlled operator equation [Formula omitted], [Formula omitted], where [Formula omitted] is the control parameter from, generally speaking,an arbitrary set [Formula omitted]. The abstract theory is illustrated by examples of acontrolled nonlinear operator differential equation in a Banach space as well as a stronglynonlinear pseudoparabolic equation. This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator acting on an arbitrary Banach space , new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used: , where is, generally speaking, another Banach space semi-ordered by the cone . These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov (, ). The role of an analog of the Lipschitz constant on a given bounded set is played by a bounded linear operator , depending on this set, with spectral radius . In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation , , where is the control parameter from, generally speaking, an arbitrary set . The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation. |
| Audience | Academic |
| Author | Chernov, A. V. |
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SuminV.I.ChernovA.V.Operators in spaces of measurable functions: Volterra and quasi-nilpotencyDiffer. Equations199834101403141117130110958.47014 Chernov, A.V., Volterra operator equations and their application in the theory of optimization of hyperbolic systems, Cand. Sci. (Phys.-Math.) Dissertation, Nizhny Novgorod, 2000. ChernovA.V.Smooth finite-dimensional approximations of distributed optimization problems via control discretizationComput. Math. Math. Phys.2013531218391852314657110.1134/S096554251312004X KalininA.V.MorozovS.F.The Cauchy problem for a nonlinear integro-differential transport equationMath. Notes1997615566573162011310.1007/BF02355077 Krasnosel’skii, M.A., Polozhitel’nye resheniya operatornykh uravnenii. Glavy nelineinogo analiza (Positive Solutions of Operator Equations. Chapters of Nonlinear Analysis), Moscow: Fizmatgiz, 1962. KobayashiT.PecherH.ShibataY.On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosityMath. Ann.19932962215234121990010.1007/BF01445103 LionsJ.-L.Controle de systèmes distribués singuliers1983ParisGauthier-Villars0514.93001 Sumin, V.I., On the problem of singularity of distributed control systems. III, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2002, no. 1 (25), pp. 164–174. Sumin, V.I., The problem of stability of the existence of global solutions of controlled boundary value problems and Volterra functional equations, Vestn. Nizhegorod. Gos. Univ. Mat., 2003, no. 1, pp. 91–107. ChernovA.V.On totally global solvability of controlled second kind operator equationVestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki202030192111411950210.35634/vm200107 ChernovA.V.The total preservation of unique global solvability of the first kind operator equation with additional controlled nonlinearityRuss. Math.20186211536610.3103/S1066369X18110063 LuG.Global existence and blow-up for a class of semilinear parabolic systems: a Cauchy problemNonlinear Anal. Theory Methods Appl.199524811931206132564210.1016/0362-546X(94)00190-S FursikovA.V.Optimal Control of Distributed Systems. Theory and Applications2000Providence, R.I.Am. Math. Soc.1027.93500 2370_CR7 2370_CR25 2370_CR9 A.V. Chernov (2370_CR22) 2017; 61 2370_CR8 A.V. Chernov (2370_CR23) 2020; 30 2370_CR27 V.I. Sumin (2370_CR21) 2019; 25. N 1 A.V. Kalinin (2370_CR2) 1997; 61 H. Saito (2370_CR31) 2018; 264 A. Tersenov (2370_CR30) 2009; 1 A.V. Chernov (2370_CR20) 2018; 58 A.V. Chernov (2370_CR32) 2018; 62 A.V. Fursikov (2370_CR6) 2000 2370_CR10 V.I. Sumin (2370_CR11) 2000; 5 2370_CR13 2370_CR12 2370_CR34 2370_CR33 G. Lu (2370_CR29) 1995; 24 2370_CR17 V.I. Sumin (2370_CR19) 2018; 51 2370_CR16 A.V. Chernov (2370_CR18) 2015; 25 M.O. Korpusov (2370_CR26) 2006; 40 V.K. Kalantarov (2370_CR24) 1978; 10 T. Kobayashi (2370_CR28) 1993; 296 M.O. Korpusov (2370_CR4) 2005; 41 V.I. Sumin (2370_CR14) 1990; 30 V.I. Sumin (2370_CR35) 1998; 34 2370_CR3 2370_CR1 J.-L. Lions (2370_CR5) 1983 A.V. Chernov (2370_CR15) 2013; 53 |
| References_xml | – reference: Sumin, V.I., Functional Volterra equations in the mathematical theory of optimal control of distributed systems, Doctoral (Phys.-Math.) Dissertation, Nizhny Novgorod, 1998. – reference: ChernovA.V.On totally global solvability of controlled second kind operator equationVestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki202030192111411950210.35634/vm200107 – reference: Sumin, V.I., On the problem of singularity of distributed control systems. II, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2001, no. 1 (23), pp. 198–204. – reference: KorpusovM.O.SveshnikovA.G.Blow-up of solutions of strongly nonlinear equations of pseudoparabolic typeSovrem. Mat. Pril.2006403138 – reference: KorpusovM.O.Global solvability conditions for an initial–boundary value problem for a nonlinear equation of pseudoparabolic typeDiffer. Equations2005415712720220068010.1007/s10625-005-0206-2 – reference: Sumin, V.I., The problem of stability of the existence of global solutions of controlled boundary value problems and Volterra functional equations, Vestn. Nizhegorod. Gos. Univ. Mat., 2003, no. 1, pp. 91–107. – reference: Sumin, V.I. and Chernov, A.V., Volterra functional-operator equations in the theory of optimization of distributed systems, Tr. Mezhdunar. konf. “Dinamika sistem i protsessy upravleniya,” posvyashch. 90-letiyu so dnya rozhd. akad. N.N. Krasovskogo (Proc. Int. Conf. “Dynamics of Systems and Control Processes” Dedicated 90th Anniv. Acad. N.N. Krasovskii), (Yekaterinburg, September 15–20, 2014), Yekaterinburg, 2015, pp. 293–300. – reference: Tröltzsch, F., Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Grad. Stud. Math., Providence, 2010, vol. 112. – reference: Gaewski, H., Gröger, K, and Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Berlin: Akademie-Verlag, 1974. – reference: Krasnosel’skii, M.A., Polozhitel’nye resheniya operatornykh uravnenii. Glavy nelineinogo analiza (Positive Solutions of Operator Equations. Chapters of Nonlinear Analysis), Moscow: Fizmatgiz, 1962. – reference: Sumin, V.I., On the problem of singularity of distributed control systems. III, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2002, no. 1 (25), pp. 164–174. – reference: Chernov, A.V., Volterra operator equations and their application in the theory of optimization of hyperbolic systems, Cand. Sci. (Phys.-Math.) Dissertation, Nizhny Novgorod, 2000. – reference: Chernov, A.V., On overcoming the singularity of distributed control systems, Tr. Tret’ei Vseross. nauchn. konf. “Matematicheskoe modelirovanie i kraevye zadachi” (Proc. Third All-Russ. Sci. 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| SubjectTerms | Banach spaces Difference and Functional Equations Differential equations Existence theorems Linear operators Lipschitz condition Mathematical analysis Mathematics Mathematics and Statistics Nonlinear control Operators (mathematics) Ordinary Differential Equations Partial Differential Equations Uniqueness |
| Title | Operator Equations of the Second Kind: Theorems on the Existence and Uniqueness of the Solution and on the Preservation of Solvability |
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