Mathematical bases of optimal measurements theory in nonstationary case

M. A. Sagadeeva

Journal of computational and engineering mathematics, Tome 3 (2016) no. 3, pp. 19-32.

Résumé

Recently, the use of mathematical results is becoming increasingly vast field of study for solving technical problems. An example of such approach is the recently developed optimal measurement theory. In the article the mathematical reasoning for solution of the measurement problem of dynamically distorted signal, taking into account the multiplier effect on the measuring transducer (MT). Making such a change can improve the adequacy of the mathematical model of the MT, namely, the problem is considered under the assumption that the MT are subject to change over time, which allows us to describe a decrease in sensitivity of elements of the MT.

Détail

Keywords: nonstationary Sobolev type equations, relatively bounded operator, degenerate flow of operators, optimal control problem, Showalter – Sidorov problem.

Bibliographie

[1] V. A. Granovskii, Dinamicheskie izmereniya. Osnovy metrologicheskogo obespecheniya, Energoatomizdat, L., 1984, 224 pp.

[2] A. L. Shestakov, Metody teorii avtomaticheskogo upravleniya v dinamicheskikh izmereniyakh, Izd. tsentr YuUrGU, Chelyabinsk, 2013, 257 pp.

[3] G. A. Sviridyuk, A. A. Efremov, “Optimalnoe upravlenie odnim klassom lineinykh vyrozhdennykh uravnenii”, Doklady Akademii nauk, 246:3 (1999), 323–325 | MR

[4] A. L. Shestakov, G. A. Sviridyuk, “Novyi podkhod k izmereniyu dinamicheski iskazhennykh signalov”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2010, no. 5, 116–120 | Zbl

[5] A. V. Keller, “Chislennoe reshenie zadachi optimalnogo upravleniya vyrozhdennoi lineinoi sistemoi uravnenii s nachalnymi usloviyami Shouoltera–Sidorova”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2008, no. 27 (127), 50–56 | Zbl

[6] A. V. Keller, Chislennoe issledovanie zadach optimalnogo upravleniya dlya modelei leontevskogo tipa, diss. \ldots dokt. fiz.-matem. nauk, Chelyabinsk, 2011, 237 pp.

[7] A. L. Shestakov, A. V. Keller, E. I. Nazarova, “Numerical Solution of the Optimal Measurement Problem”, Automation and Remote Control, 73:1 (2012), 97–104 | DOI | MR | Zbl

[8] A. L. Shestakov, A. V. Keller, G. A. Sviridyuk, “The theory of optimal measurements”, J. Comp. Eng. Math., 1:1 (2014), 3–16 | Zbl

[9] A. L. Shestakov, G. A. Sviridyuk, Yu. V. Khudyakov, “Dinamicheskie izmereniya v prostranstvakh «shumov» usloviyami Shouoltera–Sidorova”, Vestnik YuUrGU. Seriya: Kompyuternye tekhnologii, upravlenie, radioelektronika, 13:2 (2013), 4–11

[10] G. A. Sviridyuk, N. A. Manakova, “Dinamicheskie modeli sobolevskogo tipa s usloviem Shouoltera–Sidorova i additivnymi «shumami»”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 7:1 (2014), 90–103 | DOI | Zbl

[11] A. V. Keller, M. A. Sagadeeva, “Chislennoe reshenie zadach optimalnogo i zhestkogo upravleniya dlya odnoi nestatsionarnoi sistemy leontevskogo tipa”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Seriya: Matematika. Fizika, 32:19 (2013), 57–66

[12] M. A. Sagadeeva, G. A. Sviridyuk, “The Nonautonomous Linear Oskolkov Model on a Geometrical Graph: the Stability of Solutions and Optimal Control Problem”, Semigroups of Operators – Theory and Applications, Proc. Int. Conference (Bedlewo, Poland, 2013, October 6–10), Springer Proceedings in Mathematics and Statistics, 113, eds. J. Banasiak, A. Bobrowski, M. Lachowicz, Springer International Publishing, 2015, 257–271 | DOI | MR | Zbl

[13] A. L. Shestakov, G. A. Sviridyuk, M. A. Sagadeeva, “Reconstruction of a Dynamically Distorted Signal with Respect to the Measure Tranducer Degradation”, Applied Mathematical Sciences, 8:41–44 (2014), 2125–2130 | DOI

[14] A. L. Shestakov, A. V. Keller, M. A. Sagadeeva, “Numerical Algorithm for Reconstruction of a Dynamically Distorted Signal with Inertia and Multiplicative Effect”, Applied Mathematical Sciences, 8:113–116 (2014), 5731–5736 | DOI

[15] A. A. Belov, A. P. Kurdyukov, Deskriptornye sistemy i zadachi upravleniya, Fizmatlit, M., 2015, 300 pp.

[16] G. A. Sviridyuk, T. G. Sukacheva, “Phase spaces of a class of operator semilinear equations of Sobolev type”, Differ. Equ., 26:2 (1990), 188–195 | MR | Zbl

[17] G. A. Sviridyuk, T. G. Sukacheva, “Rapid-Slow Dynamics of Viscoelastic Media”, Soviet Mathematics Doklads, 40:2 (1990), 376–379 | MR | Zbl

[18] G. V. Demidenko, S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-order Derivative, Marcel Dekker Inc., New York-Basel-Hong Kong, 2003, xvi+490 pp. | MR | Zbl

[19] N. Sidorov, B. Loginov, A. Sinithyn and M. Falaleev, Lyapunov–Shmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002, xx+548 pp. | DOI | MR

[20] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston, 2003, viii+216 pp. | MR | Zbl

[21] A. A. Zamyshlyaeva, Lineinye uravneniya sobolevskogo tipa vysokogo poryadka, Izd. tsentr YuUrGU, Chelyabinsk, 2012, 107 pp. | MR

[22] N. A. Manakova, Zadachi optimalnogo upravleniya dlya polulineinykh uravnenii sobolevskogo tipa, Izd. tsentr YuUrGU, Chelyabinsk, 2012, 88 pp. | MR

[23] M. A. Sagadeeva, Dikhotomii reshenii lineinykh uravnenii sobolevskogo tipa, Izd. tsentr YuUrGU, Chelyabinsk, 2012, 139 pp. | MR

[24] O. P. Matveeva, T. G. Sukacheva, Matematicheskie modeli vyazkouprugikh neszhimaemykh zhidkostei nenulevogo poryadka, Izd. tsentr YuUrGU, Chelyabinsk, 2014, 101 pp.

[25] S. A. Zagrebina, M. A. Sagadeeva, Ustoichivye i neustoichivye mnogoobraziya reshenii polulineinykh uravnenii sobolevskogo tipa, Izd. tsentr YuUrGU, Chelyabinsk, 2016, 121 pp.

[26] A. V. Keller, Dzh. K. Al-Delfi, “Golomorfnye vyrozhdennye gruppy operatorov v kvazibanakhovykh prostranstvakh”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 7:1 (2015), 20–27 | Zbl

[27] A. A. Zamyshlyaeva, D. K. T. Al-Isawi, “On some properties of solutions to one class of evolution Sobolev type mathematical models in quasi-Sobolev spaces”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:4 (2015), 113–119 | DOI | Zbl

[28] M. A. Sagadeeva, F. L. Khasan, “Ogranichennye resheniya modeli Barenblatta–Zheltova–Kochinoi v kvazisobolevykh prostranstvakh”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 8:4 (2015), 138–144 | DOI | Zbl

[29] A. Favini, G. A. Sviridyuk, A. A. Zamyshlyaeva, “One Class of Sobolev Type Equations of Higher Order with Additive “White Noise””, Communications on Pure and Applied Analysis, 15:1 (2016), 185–196 | DOI | MR | Zbl

[30] A. Favini, G. A. Sviridyuk, N. A. Manakova, “Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of “Noises””, Abstract and Applied Analysis, 2015 (2015), 697410, 8 pp. | DOI | MR | Zbl

[31] A. L. Shestakov, G. A. Sviridyuk, Yu. V. Khudyakov, “Dynamical Measurements in the View of the Group Operators Theory”, Semigroups of Operators – Theory and Applications, Proc. Int. Conference (Bedlewo, Poland, 2013, October 6–10), Springer Proceedings in Mathematics and Statistics, 113, eds. J. Banasiak, A. Bobrowski, M. Lachowicz, Springer International Publishing, 2015, 273–286 | DOI | MR | Zbl

[32] M. A. Sagadeeva, “Vyrozhdennye potoki razreshayuschikh operatorov dlya nestatsionarnykh uravnenii sobolevskogo tipa”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 9:1 (2017) (to appear)

[33] A. D. Ioffe, V. M. Tikhomirov, Teoriya ekstremalnykh zadach, Nauka, M., 1974, 480 pp. | MR