Sufficiency of Kalman’s Rank Condition for the Approximate Boundary Controllability on an Annular Domain

Abstract

In this paper the author establishes the sufficiency of Kalman’s rank condition on the approximate boundary controllability at a finite time for diagonalizable systems on an annular domain in higher dimensional case.

Keywords

Kalman’s rank condition / Approximate boundary controllability / Diagonalizable systems of wave equations / Annular domain

References

[1] Atkinson K, Han W M. Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, 2012, Heidelberg, Tomsk: Springer-Verlag

[2] Bagrov V G, Zadorozhnyi V N, Trifonov A Yu. Methods of Mathematical Physics, Special Functions, Equations of Mathematical Physics, 2012, Tomsk: Tomsk Polytechnic University

[3] Bobkov V. Asymptotic relation for zeros of cross-product of Bessel functions and applications. J. Math. Anal. Appl., 2019, 472(1): 1078-1092

[4] Gardiner S J, Render H. Harmonic functions which vanish on coaxial cylinders. J. Anal. Math., 2019, 138(2): 891-915

[5] Li T T, Rao B P. Criteria of Kalman’s type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls. SIAM J. Control Optim., 2016, 54(1): 49-72

[6] Li T T, Rao B P. Boundary Synchronization for Hyperbolic Systems, volume 94, Progress in Nonlinear Differential Equations and their Applications, Subseries in Control, 2019, Cham: Birkhäuser/Springer-Verlag

[7] Tikhonov A N, Samarskiĭ A A. Equations of Mathematical Physics, 1990, New York: Dover Publications, Inc. Translated from the Russian by A. R. M. Robson and P. Basu, Reprint of the 1963 translation

[8] Wu C S. Methods of Mathematical Physics, 2019, Beijing: Peking University Press (in Chinese)

[9] Zu C X, Li T T, Rao B P. Sufficiency of Kalman’s rank condition for the approximate boundary controllability at finite time on a spherical domain. Math. Methods Appl. Sci., 2021, 44: 13509-13525