In this paper, we consider the controllability of a strongly degenerate parabolic equation with a degenerate one-order transport term. Despite the strong degeneracy, we prove a result of well-posedness and null controllability with a Dirichlet boundary control that acts on the degenerate part of the boundary. Then, we study the uniform controllability in the vanishing viscosity limit and prove that the cost of the control explodes exponentially fast in small time and converges exponentially fast in large time in some adapted weighted norm. The main tools used are a spectral decomposition involving Bessel functions and their zeros, some usual results on admissibility of scalar controls for diagonal semigroups, and the moment method of Fattorini and Russell.
Accepted:
DOI: 10.1051/cocv/2016036
Keywords: Degenerate parabolic equation, cost of the control, uniform controllability, Bessel functions
Gueye, Mamadou 1; Lissy, Pierre 2
@article{COCV_2016__22_4_1184_0,
author = {Gueye, Mamadou and Lissy, Pierre},
title = {Singular optimal control of a {1-D} parabolic-hyperbolic degenerate equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1184--1203},
publisher = {EDP Sciences},
volume = {22},
number = {4},
year = {2016},
doi = {10.1051/cocv/2016036},
zbl = {1357.35202},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2016036/}
}
TY - JOUR AU - Gueye, Mamadou AU - Lissy, Pierre TI - Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1184 EP - 1203 VL - 22 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2016036/ DO - 10.1051/cocv/2016036 LA - en ID - COCV_2016__22_4_1184_0 ER -
%0 Journal Article %A Gueye, Mamadou %A Lissy, Pierre %T Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1184-1203 %V 22 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2016036/ %R 10.1051/cocv/2016036 %G en %F COCV_2016__22_4_1184_0
Gueye, Mamadou; Lissy, Pierre. Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 1184-1203. doi: 10.1051/cocv/2016036
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables. Vol. 55 of Appl. Math. Series. National Bureau of Standards (1964). | Zbl
, Transport equation and Cauchy Problem for BV vector fields. Invent. Math. 158 (2004) 227–260. | Zbl | DOI
S.A. Avdonin and S.A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, Cambridge New York (1995). | Zbl
, and , Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47 (2008) 1–19. | Zbl | DOI
, and , Unique continuation and approximate controllability for a degenerate parabolic equation. Appl. Anal. 91 (2012) 1409–1425. | Zbl | DOI
and , On the non-uniform null controllability of a linear KdV equation. Asymptot. Anal. 94 (2015) 33–69. | Zbl
and , Singular optimal control: A linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237–257. | Zbl
and , Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–547. | Zbl | DOI
and , Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971) 272–292. | Zbl | DOI
A.V. Fursikov and O.Y. Imanuvilov, Controllability of Evolution Equations. Vol. 34 of Lecture Notes. Seoul National University, Korea (1996). | Zbl
, A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Anal. 258 (2010) 852–868. | Zbl | DOI
and , On the uniform controllability of the Burgers equation. SIAM J. Control Optim. 46 (2007) 1211–1238. | Zbl | DOI
and , Uniform controllability of a transport equation in zero diffusion-dispersion limit. Math. Models Methods Appl. Sci. 19 (2009) 1567–1601. | Zbl | DOI
and , Singular optimal control for a transport-diffusion equation. Commun. Partial Differ. Eq. 32 (2007) 1813–1836. | Zbl | DOI
, Exact boundary controllability of parabolic and hyperbolic equations. SIAM J. Control Optim. 52 (2014) 2037–2054. | Zbl | DOI
and , Admissible Input Elements for Systems in Hilbert Space and a Carleson Measure Criterion. SIAM J. Control Optim. 21 614–640. | Zbl
E. Kamke, Differentialgleichungen: L’sungsmethoden und l’sungen, rd edition. Chelsea Publishing Company, New York (1948). | Zbl
and , Controle exact de l’équation de la chaleur. Commun. Partial Differ. Eq. 20 (1995) 335–356. | Zbl
V. Komornik and P. Loreti, Fourier series in control theory. Springer (2005). | Zbl
, Uniform controllability of scalar conservation laws in the vanishing viscosity limit. SIAM, J. Control Optim. 50 (2012) 1661–1699. | Zbl
, A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation. C. R. Math. Acad. Sci. Paris 350 (2012) 591–595. | Zbl | MR | DOI
, An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit. Syst. Control Lett. 69 (2014) 98–102. | Zbl
, On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension. SIAM J. Control Optim. 52 (2014) 2651–2676. | Zbl
, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation. J. Differ. Equ. 259 (2015) 5331–5352. | Zbl
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Springer, Berlin (1972). | Zbl
and , Monotonic sequences related to zeros of Bessel functions. Numer. Algor. 49 (2008) 221–233. | Zbl
P. Koosis, The logarithmic integral I & II. Vol. 12 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1988) and Vol. 21 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1992). | Zbl
and , Trace Formulas for Some Singular Differential Operators and Applications. Math. Nachr. 211 (2000) 127–157. | Zbl | DOI
W. Rudin, Real and complex analysis. McGraw-Hill Book Co., New York (1966). | Zbl
H. Tanabe, Equations of evolution [English transl., Iwanami, Tokyo (1975)]. Pitman, London (1979). | Zbl
and , New blow-up rates for fast controls of Schrodinger and heat equations. J. Differ. Equ. 243 (2007) 70–100. | Zbl | DOI
G.N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, England (1958). | Zbl
Cited by Sources:
