On problems of boundary control and optimal control of a distributed inhomogeneous oscillatory system with given intermediate conditions on the state functions

V. R. Barseghyan

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 13-29.

Résumé

In this work, we examine a distributed inhomogeneous oscillatory system, in which various states are specified at intermediate times. Problems of boundary control and optimal boundary control of this system are considered. The dynamics of this object is modeled by a one-dimensional wave equation with piecewise constant characteristics; the oscillations propagate in homogeneous domains areas in the same time. The quality criterion for optimal boundary control problems is specified over the entire time interval. A constructive approach to constructing a boundary control function and optimal control of one-dimensional oscillatory inhomogeneous processes is proposed. The research approach is based on methods of separation of variables, control theory, and optimal control of finite-dimensional systems with multipoint intermediate conditions. Under the influence of the constructed control law, wave oscillations from a given initial state pass into a given terminal state through multipoint intermediate states.

Détail

Keywords: boundary control of oscillations, optimal control of oscillations, inhomogeneous oscillatory process, wave equation, piecewise constant characteristics

Bibliographie

[1] Barsegyan V. R., “Optimalnoe granichnoe upravlenie smescheniem na dvukh kontsakh pri kolebanii sterzhnya, sostoyaschego iz dvukh uchastkov raznoi plotnosti i uprugosti”, Differ. uravn. protsessy upravl., 2022, no. 2, 41–54 | Zbl

[2] Barsegyan V. R., “Optimalnoe granichnoe upravlenie raspredelennoi neodnorodnoi kolebatelnoi sistemoi s zadannymi sostoyaniyami v promezhutochnye momenty vremeni”, Zh. vychisl. mat. mat. fiz., 63:1 (2023), 74–84 | DOI

[3] Butkovskii A. G., metody upravleniya sistemami s raspredelennymi parametrami, Nauka, M., 1975

[4] Egorov A. I., Znamenskaya L. N., “Ob upravlyaemosti uprugikh kolebanii posledovatelno soedinennykh ob'ektov s raspredelennymi parametrami”, Tr. In-ta mat. mekh. UrO RAN., 17:1 (2011), 85–92

[5] Zvereva M. B., Naidyuk F. O., Zalukaeva Zh. O., “Modelirovanie kolebanii singulyarnoi struny”, Vestn. Voronezh. gos. un-ta. Ser. Fiz. Mat., 2014, no. 2, 111–119 | Zbl

[6] Zubov V. I., Lektsii po teorii upravleniya, Nauka, M., 1975

[7] Ilin V. A., “Optimizatsiya granichnogo upravleniya kolebaniyami sterzhnya, sostoyaschego iz dvukh raznorodnykh uchastkov”, Dokl. RAN., 440:2 (2011), 159–163 | Zbl

[8] Ilin V. A., “O privedenii v proizvolno zadannoe sostoyanie kolebanii pervonachalno pokoyaschegosya sterzhnya, sostoyaschego iz dvukh raznorodnykh uchastkov”, Dokl. RAN., 435:6 (2010), 732–735 | Zbl

[9] Korzyuk V. I., Kozlovskaya I. S., “Dvukhtochechnaya granichnaya zadacha dlya uravneniya kolebaniya struny s zadannoi skorostyu v nekotoryi moment vremeni. II”, Tr. In-ta mat. NAN Belarusi., 19:1 (2011), 62–70 | MR | Zbl

[10] Krasovskii N. N., Teoriya upravleniya dvizheniem, Nauka, M., 1968

[11] Kuleshov A. A., “Smeshannye zadachi dlya uravneniya prodolnykh kolebanii neodnorodnogo sterzhnya i uravneniya poperechnykh kolebanii neodnorodnoi struny, sostoyaschikh iz dvukh uchastkov raznoi plotnosti i uprugosti”, Dokl. RAN., 442:5 (2012), 594–597 | Zbl

[12] Lvova N. N., “Optimalnoe upravlenie nekotoroi raspredelennoi neodnorodnoi kolebatelnoi sistemoi”, Avtomat. telemekh., 1973, no. 10, 22–32 | Zbl

[13] Provotorov V. V., “Postroenie granichnykh upravlenii v zadache o gashenii kolebanii sistemy strun”, Vestn. Sankt-Peterb. un-ta. Prikl. mat. Inform. Protsessy upravl., 1 (2012), 62–71

[14] Rogozhnikov A. M., “Issledovanie smeshannoi zadachi, opisyvayuschei protsess kolebanii sterzhnya, sostoyaschego iz neskolkikh uchastkov s proizvolnymi dlinami”, Dokl. RAN., 444 (2012), 488–491 | MR | Zbl

[15] Amara J. Ben, Beldi E., “Boundary controllability of two vibrating strings connected by a point mass with variable coefficients”, SIAM J. Control Optim., 57:5 (2019), 3360–3387 | DOI | MR | Zbl

[16] Amara J. Ben, Bouzidi H., “Null boundary controllability of a one-dimensional heat equation with an internal point mass and variable coefficients”, J. Math. Phys., 59:1 (2018), 1–22 | MR | Zbl

[17] Barseghyan V. R., “Control problem of string vibrations with inseparable multipoint conditions at intermediate points in time”, Mech. Solids., 54:8 (2019), 1216–1226 | DOI

[18] Barseghyan V. R., “The problem of boundary control of displacement at two ends by the process of oscillation of a rod consisting of two sections of different density and elasticity”, Mech. Solids., 58:2 (2023), 483–491 | DOI | MR | Zbl

[19] Barseghyan V. R., “On the controllability and observability of linear dynamic systems with variable structure”, Proc. Int. Conf. “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy's Conference) (Moscow, June 1-3, 2016), 2016, 1–3

[20] Barseghyan V. R., Barseghyan T. V., “On an approach to the problems of control of dynamic system with nonseparated multipoint intermediate conditions”, Automat. Remote Control., 76:4 (2015), 549–559 | DOI | MR | Zbl

[21] Barseghyan V., Solodusha S., “On the optimal control problem for vibrations of the rod/string consisting of two non-homogeneous sections with the condition at an intermediate iime”, Mathematics., 10:23 (2022), 4444 | DOI

[22] Barseghyan V., Solodusha S., “Control of string vibrations by displacement of one end with the other end fixed, given the deflection form at an intermediate moment of time”, Axioms., 11:4 (2022), 157 | DOI | MR

[23] Barseghyan V., Solodusha S., “Optimal boundary control of string vibrations with given shape of deflection at a certain moment of time”, Lect. Notes Comp. Sci., 12755 (2021), 299–313 | DOI | MR | Zbl

[24] Barseghyan V., Solodusha S., “On one problem in optimal boundary control for string vibrations with a given velocity of points at an intermediate moment of time”, Proc. International Russian Automation Conference (RusAutoCon) (Sochi, September 5-11, 2021), Sochi, 2021, 343–349 | MR

[25] Barseghyan V. R., Solodusha S. V., “On one boundary control problem of string vibrations with given velocity of points at an intermediate moment of time”, J. Phys. Conf. Ser., 2021 | MR

[26] Mercier D., Regnier V., “Boundary controllability of a chain of serially connected Euler–Bernoulli beams with interior masses”, Collect. Math., 60:3 (2009), 307–334 | DOI | MR | Zbl