Geodesic
The theory of optimal measurements
Journal of computational and engineering mathematics, Tome 1 (2014) no. 1, pp. 3-16.

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The mathematical model (MM) of the measuring transducer (MT) is discussed. The MM is intended for restoration of deterministic signals distorted by mechanical inertia of the MT, resonances in MT's circuits and stochastic perturbations. The MM is represented by the Leontieff type system of equations, reflecting the change in the state of MT under useful signal, deterministic and stochastic perturbations; algebraic system of equations modelling observations of distorted signal; and the Showalter – Sidorov initial condition. In addition the MM of the MT includes a cost functional. The minimum point of a cost functional is a required optimal measurement. Qualitative research of the MM of the MT is conducted by the methods of the degenerate operator group's theory. Namely, the existence of the unique optimal measurement is proved. This result corresponds to input signal without stochastic perturbation. To consider stochastic perturbations it is necessary to introduce so called Nelson – Gliklikh derivative for random process. In conclusion of article observations of "noises" (random perturbation, especially "white noise") are under consideration.
Keywords: mathematical model of the measuring transducer, the Leontieff type system, the Showalter – Sidorov condition, cost functional, the Nelson – Gliklikh derivative, «white noise». 
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A. L. Shestakov; A. V. Keller; G. A. Sviridyuk. The theory of optimal measurements. Journal of computational and engineering mathematics, Tome 1 (2014) no. 1, pp. 3-16. http://geodesic.mathdoc.fr/item/JCEM_2014_1_1_a0/

[1] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992, 476 pp. | MR | Zbl

[2] F. R. Gantmacher, The Theory of Matrices, AMS Chelsea Publishing, Providence, 2000, 626 pp. | MR

[3] Yu. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics, Springer, London, 2011 | DOI | MR | Zbl

[4] Yu. E. Gliklikh, “Investigation of Leontieff Type Equations with White Noise by the Methods of Mean Derivatives of Stochastic Processes”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 13, 24–34 | Zbl

[5] A. V. Keller, “Numerical Solution of the Optimal Control Problem for Degenerate Linear System of Equations with Showalter–Sidorov Initial Conditions”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2008, no. 27 (127), 50–56 | Zbl

[6] A. V. Keller, “The Leontief type systems: classes of problems with the Showalter–Sidorov intial condition and numerical solving”, IIGU Ser. Matematika, 3:2 (2010), 30–43 | Zbl

[7] A. V. Keller, Numerical Study of Optimal Control Problems for Leontieff Type Models, Diss. ... doct. phys.-math. sciences, Chelyabinsk, 2011, 237 pp.

[8] A. V. Keller, E. I. Nazarova, “The regularization property and the computational solution of the dynamic measure problem”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2010, no. 5, 32–38 | Zbl

[9] Yu. V. Khudyakov, “The Numerical Algorithm to Study the Shestakov-Sviridyuk Model of the Measuring Device with Inertia and Resonances”, Matematicheskiye Zametki SVFU, 20:2 (2013), 211–221 | Zbl

[10] M. Kovács, S. Larsson, “Introduction to Stochastic Partial Differential Equations”, Processing of «New Directions in the Mathematical and Computer Sciences» (National Universities Commission, Abuja, Nigeria, October 8-12, 2007), Publications of the ICMCS, 4, 2008, 159–232 | Zbl

[11] N. A. Manakova, “On a hyposithis of G. A. Sviridyuk”, IIGU Ser. Matematika, 4:4 (2011), 87–93 | Zbl

[12] I. V. Melnikova, A. I. Filinkov, M. A. Alshansky, “Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions”, Journal of Mathematical Sciences, 116:5 (2003), 3620–3656 | DOI | MR | Zbl

[13] I. V. Melnikova, A. I. Filinkov, M. A. Alshansky, “Generalized solutions to abstract stochastic problems”, J. Integ. Transf. and Special Funct., 20:3-4 (2009), 199–206 | DOI | MR | Zbl

[14] E. Nelson, Dynamical Theory of Brownian Motion, Princeton University Press, Princeton, 1967, 142 pp. | MR

[15] A. L. Shestakov, “Dynamic Accuracy of the Transmitter With a Correction Device as a Sensor Model”, Metrology, 1987, no. 2, 26–34 (in Russian) | MR

[16] 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

[17] A. L. Shestakov, G. A. Sviridyuk, “A new approach to measurement of dynamically perturbed signals”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2010, no. 5, 116–120 | Zbl

[18] A. L. Shestakov, G. A. Sviridyuk, “Optimal measurement of dynamically distorted signals”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2011, no. 8, 70–75 | Zbl

[19] A. L. Shestakov, G. A. Sviridyuk, “On a New Conception of White Noise”, Obozrenie Prikladnoy i Promyshlennoy Matematiki, 19:2 (2012), 287–288

[20] A. L. Shestakov, G. A. Sviridyuk, “On the Measurement of the «White Noise»”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 13, 99–108 | Zbl

[21] A. L. Shestakov, G. A. Sviridyuk, Yu. V. Khudyakov, “Dynamic Measurement in Spaces of «Noise»”, Bulletin of the South Ural State University. Series «Computer Technologies, Automatic Control Radioelectronics», 13:2 (2013), 4–11

[22] 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

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

[24] G. A. Sviridyuk, N. A. Manakova, “The Dynamical Models of Sobolev Type with Showalter–Sidorov Condition and Additive “Noise””, Vestnik YuUrGU. Ser. Mat. Model. Progr., 7:1 (2014), 90–103 | DOI | Zbl

[25] G. A. Sviridyuk, S. A. Zagrebina, “The Showalter–Sidorov problem as a phenomena of the Sobolev-type equations”, IIGU Ser. Matematika, 3:1 (2010), 104–125 | MR | Zbl

[26] S. A. Zagrebina, E. A. Soldatova, “The linear Sobolev-type Equations With Relatively $p$-bounded Operators and Additive White Noise”, IIGU Ser. Matematika, 6:1 (2013), 20–34 | Zbl

[27] A. A. Zamyshlyaeva, “Stochastic Incomplete Linear Sobolev Type High-Ordered Equations with Additive White Noise”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 14, 73–82 | MR | Zbl