Biorthogonal Functions for Complex Exponentials and an Application to the Controllability of the Kawahara Equation Via a Moment Approach
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Avdonin, S.A., Ivanov, S.A.: Families of exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press, Cambridge (1995)
Bugariu, I.F., Micu, S.: A singular controllability problem with vanishing viscosity. ESAIM Control Optim. Calc. Var. 20, 116–140 (2014)
Capistrano-Filho, R.A., Chentouf, B., de Sousa, L.S., Gonzalez Martinez, V.H.: Two stability results for the Kawahara equation with a time-delayed boundary control. Z. Angew. Math. Phys. 74(16), 26 (2023)
Capistrano-Filho, R.A., Gomes, M.M.S.: Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces. Nonlinear Anal. 207, 112267 (2021)
Cerpa, E.: Control of a Korteweg-de Vries equation: a tutorial. Math. Control Relat. Fields 4, 45–99 (2014)
Chen, M.: Internal controllability of the Kawahara equation on a bounded domain. Nonlinear Anal. 185, 356–373 (2019)
Fattorini, H.O., Russell, D.L.: Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quart. Appl. Math. 32, 45–69 (1974/75)
Fattorini, H.O., Russell, D.L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43, 272–292 (1971)
Flores, C., Smith, L.D.: Control and stabilization of the periodic fifth order Korteweg-de Vries equation. ESAIM Control Optim. Calc. Var. 25(38), 28 (2019)
Glass, O.: A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Anal. 258, 852–868 (2010)
Hirayama, H.: Local well-posedness for the periodic higher order KdV type equations. NoDEA Nonlinear Diff. Equ. 19, 677–693 (2012)
Laurent, C., Rosier, L., Zhang, B.-Y.: Control and stabilization of the Korteweg-de Vries equation on a periodic domain. Commun. Partial Diff. Equ. 35, 707–744 (2010)
Micu, S., de Teresa, L.: A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle. Asymptot. Anal. 66, 139–160 (2010)
Micu, S., Ortega, J.H., Pazoto, A.F.: Null-controllability of a hyperbolic equation as singular limit of parabolic ones. J. Fourier Anal. Appl. 17, 991–1007 (2011)
Paley, R.E.A.C., Wiener, N.: Fourier transforms in complex domains, vol. 19. AMS Colloq. Publ. Amer. Math. Soc., New-York (1934)
Rosier, L.: Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 2, 33–55 (1997)
Rosier, L., Zhang, B.-Y.: Control and stabilization of the Korteweg-de Vries equation: recent progresses. J. Syst. Sci. Complex. 22, 647–682 (2009)
Rosier, L., Zhang, B.-Y.: Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain. J. Differ. Equ. 254, 141–178 (2013)
Russell, D.L., Zhang, B.-Y.: Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31, 659–676 (1993)
Yan, W., Li, Y., Yang, X.: The Cauchy problem for the modified Kawahara equation in Sobolev spaces with low regularity. Math. Comput. Modell. 54, 1252–1261 (2011)
Young, R.M.: An introduction to nonharmonic Fourier series. Academic Press, New-York (1980)
Zabczyk, J.: Mathematical control theory: an introduction. Birkhuser, Basel (1992)
Zhang, B.-Y., Zhao, X.: Control and stabilization of the Kawahara equation on a periodic domain. Comm. Inf. Syst. 12, 77–96 (2012)