On the boundary control problem for  a pseudo-parabolic equation with involution:

Farrukh N. Dekhkonov

doi:10.21638/spbu10.2024.309

Authors

Farrukh N. Dekhkonov Namangan State University, 316, Uychi, Namangan, 160136, Uzbekistan https://orcid.org/0000-0003-4747-8557

DOI:

https://doi.org/10.21638/spbu10.2024.309

Abstract

Previously, some control problems for the pseudo-parabolic equation independent of involution were considered. In this paper, we consider a boundary control problem associated with a pseudo-parabolic equation with involution in a bounded one-dimensional domain. On the part of the border of the considered domain, the value of the solution with control function is given. Restrictions on the control are given in such a way that the average value of the solution in the considered domain gets a given value. The problem given by the method of separation of variables is reduced to the Volterra integral equation of the second kind. The existence of the control function was proved by the Laplace transform method.

Keywords:

boundary problem, Volterra integral equation, control function, Laplace transform, involution

Downloads

Download data is not yet available.

References

References

Cabada A., Tojo F. A. F. General results for differential equations with involutions. Differential Equations with Involutions. Paris, Atlantis Press, 2015, pp. 17–23. http://dx.doi.org/10.2991/978-94-6239-121-5-2

Carleman T. Sur la theorie des equations integrales et ses applications. Verhandl des internat Mathem. Kongr. I. Zurich, 1932, pp. 138–151.

Wiener J. Generalized solutions of functional-dierential equations. New Jersey, World Scientic Publishing, 1993, 424 p.

Dekhkonov F. N. On a boundary control problem for a pseudo-parabolic equation. Communications in Analysis and Mechanics, 2023, vol. 15, iss. 2, pp. 289–299. https://doi.org/10.3934/cam.2023015

Fayazova Z. K. Boundary control for a psevdo-parabolic equation. Mathematical Notes of NEFU, 2018, vol. 12, pp. 40–45. https://doi.org/10.25587/SVFU.2019.20.57.008

Coleman B. D., Noll W. An approximation theorem for functionals, with applications in continuum mechanics. Archive for Rational Mechanics and Analysis, 1960, vol. 6, pp. 355–370. https://doi.org/10.1007/BF00276168

Chen P., Gurtin M. On a theory of heat conduction involving two temperatures. Journal of Applied Mathematics and Physics (ZAMP), 1968, vol. 19, pp. 614–627. https://doi.org/10.1007/BF01594969

Miline E. A. The diffusion of imprisoned radiation through a gas. Journal of the London Mathematical Society, 1926, vol. 1, pp. 40–51.

White L. W. Controllability properties of pseudo-parabolic boundary control problems. SIAM Journal on Control and Optimization, 1980, vol. 18, pp. 534–539. https://doi.org/10.1137/0318039

Coleman B. D., Duffin R. J., Mizel V. J. Instability, uniqueness, and nonexistence theorems for the equation on a strip. Archive for Rational Mechanics and Analysis, 1965, vol. 19, pp. 100–116. https://doi.org/10.1007/BF00282277

White L. W. Point control of pseudo-parabolic problems. Journal of Differential Equations, 1981, vol. 42, pp. 366–374. https://doi.org/10.1016/0022-0396(81)90110-8

Lyashko S. I. On the solvability of pseudo-parabolic equations. Soviet Mathematics, 1985, vol. 29, pp. 99–101.

Fattorini H. O. Time-optimal control of solutions of operational differential equations. SIAM Journal on Control and Optimization, 1964, vol. 2, pp. 49–65.

Friedman A. Optimal control for parabolic equations. Journal of Mathematical Analysis and Applications, 1967, vol. 18, pp. 479–491. https://doi.org/10.1016/0022-247X(67)90040-6

Egorov Yu. V. Optimal'noe upravlenie v banakhovom prostranstve [Optimal control in Banach spaces]. Doklady Akademii nauk SSSR, 1963, vol. 150, pp. 241–244. (In Russian)

Albeverio S., Alimov Sh. A. On one time-optimal control problem associated with the heat exchange process. Applied Mathematics and Optimization, 2008, vol. 57, pp. 58–68. https://doi.org/10.1007/s00245-007-9008-7

Dekhkonov F. N. On the control problem associated with the heating process. Mathematical notes of NEFU, 2022, vol. 29, iss. 4, pp. 62–71. https://doi.org/10.25587/SVFU.2023.82.41.005

Fayazova Z. K. Boundary control of the heat transfer process in the space. Russian Mathematics, 2019, vol. 63, pp. 71–79. https://doi.org/10.3103/S1066369X19120089

Dekhkonov F. N., Kuchkorov E. I. On the time-optimal control problem associated with the heating process of a thin rod. Lobachevskii Journal of Mathematics, 2023, vol. 44, iss. 3, pp. 1134–1144. https://doi.org/10.1134/S1995080223030101

Dekhkonov F. N. Boundary control problem for the heat transfer equation associated with heating process of a rod. Bulletin of the Karaganda University. Mathematics Series, 2023, vol. 110, iss. 2, pp. 63–71. https://doi.org/10.31489/2023m2/63-71

Dekhkonov F. N. On the time-optimal control problem for a heat equation. Bulletin of the Karaganda University. Mathematics Series, 2023, vol. 111, iss. 3, pp. 28–38. https://doi.org/10.31489/2023m3/28-38

Lions J. L. Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Paris, Dunod Gauthier-Villars Publ., 1968, 426 p.

Fursikov A. V. Optimal control of distributed systems, theory and applications. Translations of Math. Monographs. Providence, Rhode Island, American Mathematical Society Publ., 2000, 305 p.

Altmüller A., Grüne L. Distributed and boundary model predictive control for the heat equation. GAMM-Mitteilungen, 2012, vol. 35, pp. 131–145. https://doi.org/10.1002/gamm.201210010

Dubljevic S., Christofides P. D. Predictive control of parabolic PDEs with boundary control actuation. Chem. Eng. Sci., 2006, vol. 61, pp. 6239–6248. https://doi.org/10.1016/j.ces.2006.05.041

Kritskov L. V., Sarsenbi A. M. Riesz basis property of system of root functions of second-order differential operator with involution. Differential Equations, 2017, vol. 53, pp. 33–46. https://doi.org/10.1134/S0012266117010049

Burlutskaya M. S., Khromov A. P. Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution. Comput. Math. Math. Phys., 2011, vol. 51, pp. 2102–2114. https://doi.org/10.1134/S0965542511120086

Mussirepova E., Sarsenbi A., Sarsenbi A. The inverse problem for the heat equation with reflection of the argument and with a complex coefficient. Boundary Value Problems, 2022, vol. 1, art. no. 99. https://doi.org/10.1186/s13661-022-01675-1

Kopzhassarova A., Sarsenbi A. Basis properties of eigenfunctions of second-order differential operators with involution. Abstract and Applied Analysis, 2012, vol. 2012, art. no. 576843, pp. 1–6. https://doi.org/10.1155/2012/576843

Ahmad B., Alsaedi A., Kirane M., Tapdigoglu R. An inverse problem for space and time fractional evolution equations with an involution perturbation. Quaestiones Mathematicae, 2017, vol. 40, pp. 151–160.

Dekhkonov F. N. On the control problem associated with a pseudo-parabolic type equation in an one-dimensional domain. International Journal of Applied Mathematics, 2024, vol. 37, iss. 1, pp. 109–118. https://dx.doi.org/10.12732/ijam.v37i1.9

Tikhonov A. N., Samarsky A. A. Uravneniia matematicheskoi fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 1966, 742 p. (In Russian)