Pointwise Controllability for Degenerate Parabolic Equations by the Moment Method
Tóm tắt
In this paper, we study the pointwise controllability of the one-dimensional degenerate heat equation. Necessary and sufficient conditions for approximate and null controllability are proved. Our approach is mainly based on the moment method developed by Fattorini and Russell.
Tài liệu tham khảo
Alabau-Boussouira F, Cannarsa P, Fragnelli G. Carleman estimates for degenerate parabolic operators with applications to null controllability. J Evol Equ 2006; 6:161–204.
Ammar-Khodja F, Benabdallah A, González-Burgos M., de Teresa L. Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences. J Funct Anal 2014;267(7):2077–2151.
Campiti M, Metafune G, Pallara D. Degenerate self-adjoint evolution equations on the unit interval. Semigroup Forum 1998;57:1–36.
Cannarsa P, Fragnelli G. Null controllability of semilinear degenerate parabolic equations in bounded domains. Electronic Journal of Differential Equations 2006;136:1–20.
Cannarsa P, Martinez P, Vancostenoble J. The cost of controlling strongly degenerate parabolic equations, ESAIM Control Optim. Calc. Var., to appear.
Cannarsa P, Martinez P, Vancostenoble J. The cost of controlling weakly degenerate parabolic equations by boundary controls. Mat Control Relat Fields 2017;7: 171–211.
Cannarsa P, Martinez P, Vancostenoble J. Carleman estimates for a class of degenerate parabolic operators. SIAM J Control Optim 2008;47:1–19.
Cannarsa P, Tort J, Yamamoto M. Unique continuation and approximate controllability for a degenerate parabolic equation. Appl Anal 2012;91:1409–1425.
Fernández-Cara E, González-Burgos M, de Teresa L. Boundary controllability of parabolic coupled equations. J Funct Anal 2010;259(7):1720–1758.
Coron J-M, Vol. 136. Control and nonlinearity mathematical surveys and monographs. Providence: American Mathematical Society; 2007.
Dolecki S. Observability for the one-dimensional heat equation. Studia Math 1973; 48:291–305.
Elbert A. Some recent results on the zeros of Bessel functions and orthogonal polynomials. J Comput Appl 2001;133:65–83.
Fattorini HO, Russell DL. Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quart Appl Math 1974/75;32 :45–69.
Fattorini HO, Russell DL. Exact controllability theorems for linear parabolic equations in one space dimension. Arch Ration Mech Anal 1971;43:272–292.
Gueye M. Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations. SIAM J Control Optim 2014;52:2037–2054.
Heil C. A basis theory primer, expanded edition. Applied and numerical harmonic analysis. New York: Birkhäuser/Springer; 2011.
Komornik V, Loreti P. Fourier series in control theory. Berlin: Springer; 2005.
Lions JL. Optimal control of systems governed by partial differential equations. Berlin: Springer; 1971.
Lissy P. The cost of the control in the case of a minimal time of control: the example of the one-dimensional heat equation. J Math Anal Appl 2017;451(1):497–507.
Lorch L, Muldoon ME. Monotonic sequences related to zeros of Bessel functions. Numer Algor 2008;49:221–233.
Moyano I. Flatness for a strongly degenerate 1-D parabolic equation. Math Control Signals Syst 2016;28:1–22.
Tort J, Vancostenoble J. Determination of the insolation function in the nonlinear Sellers climate model. Ann Henri Poincaré 2012;29:638–713.