Menu: Home :: go to Journal :: switch to Russian :: switch to English
You are here: all Journals and Issues→ Journal→ Issue→ Article

On the wave equation with the hysteresis type condition

Annotation

In this paper we investigate the initial-boundary value problem describing the oscillation process with a hysteresis-type boundary condition. This kind of problem arises in modeling of the string oscillations, where the movement is restricted by a sleeve concentrated at one point x = l: We suppose that the string is located along the segment [0; l] and the sleeve can move in perpendicular to [0; l] direction. The analog of d’Alembert formula is obtained. A boundary control problem is analyzed for a small period of time. The boundary control problem is to find a control function allowing to put the oscillation process from the initial state to the given final state.

Keywords

wave equation; string oscillations; d’Alembert formula; boundary control problem

Full-text in one file

Download

DOI

10.20310/1810-0198-2018-23-122-235-242

UDC

517.977

Pages

235-242

References

1. Ilin V.A., Moiseev E.I. Optimizatsiya granichnykh upravleniy kolebaniyami struny [Optimization of boundary controls of string vibrations]. Uspekhi matematicheskikh nauk – Russian Mathematical Surveys, 2005, vol. 60, no. 6 (366), pp. 89-114. (In Russian). 2. Izbrannye trudy V.A. Il'ina [Selected Works of V.A. Il’in]. Moscow, MAKS Press Publ., 2008, vol. 2, 692 p. (In Russian). 3. Egorov A.I., Znamenskaya L.N. Ob upravlyaemosti uprugikh kolebaniy posledovatel'no soedinennykh ob"ektov s raspredelennymi parametrami [On the controllability of elastic oscillations connected in series objects with distributed parameters]. Trudy Instituta matematiki i mekhaniki Ural'skogo otdeleniya RAN – Proceedings of the Steklov Institute of Mathematics, 2011, vol. 17, no. 1, pp. 85-92. (In Russian). 4. Borovskikh A.V. Formuly granichnogo upravleniya neodnorodnoy strunoy. I. [Formulas of boundary control of an inhomogeneous string: I]. Differentsial'nye uravneniya – Differential Equations, 2007, vol. 43, no. 1, pp. 64-89. (In Russian). 5. Adam L., Outrata J. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete Contin. Dyn. Syst., 2014, vol. 19, no. 9, pp. 2709-2738. 6. Adly S., Le B. K. Unbounded second-order state-dependent Moreau’s sweeping processes in Hilbert spaces. J. Optim. Theory Appl., 2016, vol. 169, no. 2, pp. 407-423. 7. Castaing C., Monteiro Marques M. BV periodic solutions of an evolution problem associated with continuous moving convex sets. Set-Valued Anal., 1995, vol. 3, no. 4, pp. 381-399. 8. Edmond J. F., Thibault L. Relaxation of an optimal control problem involving a perturbed sweeping process. Math. Program., 2005, vol. 104, no. 2-3, pp. 347-373. 9. Kamenskii M., Makarenkov O. On the response of autonomous sweeping processes to periodic perturbations. Set-Valued and Variational Analysis, 2000, vol. 24, no. 4, pp. 551-563. 10. Kamenskii M., Wen Ch.-F., Zvereva M. A string oscillations simulation with boundary conditions of hysteresis type. Optimization, 2017. DOI: https://doi.org/10.1080/02331934.2017.1388379. 11. Zvereva M. A string oscillations simulation with nonlinear conditions. Memoirs on Differential Equations and Mathematical Physics, 2017, vol. 72, pp. 141-150. 12. Zvereva M.B., Kamenskiy M.I., Shabrov S.A. Matematicheskaya model' kolebaniy struny s nelineynym usloviem [A mathematical model of string oscillations with nonlinear condition]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika – Proceedings of Voronezh State University. Series: Physics. Mathematics, 2017, no. 4, pp. 88-98. 13. Kunze M., Monteiro Marques M. An introduction to Moreau’s sweeping process. LNP, 2000, vol. 551, pp. 1-60. (In Russian). 14. Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of Mathematical Physics]. Moscow, Moscow State University Publ., 1999, 797 p. (In Russian).

Received

2018-03-27

Section of issue

Scientific articles

Для корректной работы сайта используйте один из современных браузеров. Например, Firefox 55, Chrome 60 или более новые.