Boundary control for pseudoparabolic equations in space

  • Fayazova Zarina, Department of Higher Mathematics, Tashkent State Technical University 2A Almazarskaya Street, Tashkent 100095, Uzbekistan
Keywords: pseudoparabolic equation, boundary control, admissible control, integral equation, Laplace transform


Let $u(x, y, t)$ be a solution to the pseudoparabolic equation that satisfies the initial and boundary conditions. The value of the solution is given on the part of boundary of the considered region which contains the control parameter. It is required to choose the control parameter so that on a part of the regularity domain the solution takes the specified mean value. First, we consider an auxiliary boundary value problem for a pseudoparabolic equation. We prove the existence and uniqueness of the generalized solution from the corresponding class. The restriction for the admissible control is given in the integral form. By the separation variables method, the desired problem is reduced to the Volterra integral equation. The latter is solved by the Laplace transform method. The theorem on the existence of an admissible control is proved.


Coleman B. D. and Noll W., “An approximation theorem for functionals, with applications in continuum mechanics,” Arch. Rat. Mech. Anal., 6, 355–370 (1960).

Coleman B. D., Duffin R. J., and Mizel V. J., “Instability, uniqueness, and nonexistence theorems for the equation on a strip,” Arch. Rat. Mech. Anal., 19, No. 2, 100–116 (1965).

Egorov I. E. and Efimova E. S., “Boundary value problem for third order equation not presented with respect to the highest derivative,” Mat. Zamet., 24, No. 4, 28–36 (2017).

Kozhanov A. I., “The existence of regular solutions of the first boundary value problem for one class of Sobolev type equations with alternating time direction,” Mat. Zamet. YaGU, 4, No. 2, 39–48 (1997).

White L. W., “Point control; approximations of parabolic problems and pseudo-parabolic problems,” Appl. Anal., 12, 251–263 (1981).

Lyashko S. I., “On the solvability of pseudo-parabolic equations [in Russian],” Sov. Math., 29, No. 9, 99–101 (1985).

Lyashko S. I. and Mankovskii A. A., “Optimal impulse control of a system hyperbolic type with distributed parameters [in Russian],” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 4, 69–71 (1983).

Lyashko S. I. and Mankovskii A. A., “Simultaneous optimization of momentum moments and pulse intensities in control problems for parabolic equations [in Russian],” Cybern., No. 3, 81–82, 87 (1983).

Fursikov A. V., “Feedback stabilization for 2D-seen equations: additional remarks,” in: Control and Estimation of Distributed Parameter Systems, pp. 169–187, Birkhauser, Basel (2002) (Int. Ser. Numer. Math.; 143).

Lions J.-L., Controle Optimal de Systemes Gouvernes par des Equations aux Derivees Partielles, Dunod Gauthier-Villars, Paris (1968).

Il’in V. A., “Boundary control by the oscillation process at two ends in terms of the generalized solution of the wave equation with finite energy, Differ. Equ., 36, No. 11, 1513–1528 (2000).

Il’in V. A. and Moiseev E. I., “Optimization of boundary controls by oscillations strings [in Russian],” Usp. Mat. Nauk, 60, No. 6, 89–114 (2005).

Albeverio S. and Alimov Sh., “On a time-optimal control problem associated with the heat exchange process,” Appl. Math. Optim., 57, No. 1, 58–68 (2008).

Barbu V., Ra¸scanu A., and Tessitore G., “Carleman estimates and controllability of linear stochastic heat equation,” Appl. Math. Optim., 47, No. 2, 97–120 (2003).

Fattorini H. O., “Time and norm optimal control for linear parabolic equations: necessary and sufficient conditions,” in: Control and Estimation of Distributed Parameter System, pp. 151–168, Birkhauser, Basel (2002) (Int. Ser. Numer. Math.; 143).

Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Parabolic Type Equations [in Russian], Nauka, Moscow (1967).
How to Cite
Fayazova, Z. (2019) “Boundary control for pseudoparabolic equations in space”, Mathematical notes of NEFU, 26(3), pp. 98-108. doi: