Free Access
Volume 26, 2020
Article Number 2
Number of page(s) 50
Published online 13 January 2020
  1. F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ. 6 (2006) 161–204. [CrossRef] [MathSciNet] [Google Scholar]
  2. F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences. J. Funct. Anal. 267 (2014) 2077–2151. [Google Scholar]
  3. K. Beauchard, P. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two. J. Eur. Math. Soc. 16 (2014) 67–101. [CrossRef] [MathSciNet] [Google Scholar]
  4. A. Benabdallah, F. Boyer, M. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the N-dimensional boundary null controllability in cylindrical domains. SIAM J. Control Optim. 52 (2014) 2970–3001. [Google Scholar]
  5. A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite-dimensional systems. Vol. 1 of Systems and Control: Foundations and Applications, Birkhauser Boston, Boston (1992). [Google Scholar]
  6. M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval. Semigroup Forum 57 (1998) 1–36. [CrossRef] [MathSciNet] [Google Scholar]
  7. P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift. Netw. Heterog. Media 2 (2007) 695–715. [CrossRef] [MathSciNet] [Google Scholar]
  8. P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations. Adv. Differ. Equ. 10 (2005) 153–190. [Google Scholar]
  9. P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47 (2008) 1–19. [Google Scholar]
  10. P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications. Memoirs of the American Mathematical Society. AMS (2016). [Google Scholar]
  11. P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls. Math. Control Relat. Fields 7 (2017) 171–211. [CrossRef] [Google Scholar]
  12. P. Cannarsa, P. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions. Preprint arXiv:1706.02435 (2017). [Google Scholar]
  13. P. Cannarsa, D. Rocchetti and J. Vancostenoble, Generation of analytic semi-groups in L2 for a class of second order degenerate elliptic operators. Control Cybern 37 (2008) 831–878. [Google Scholar]
  14. P. Cannarsa, J. Tort and M. Yamamoto, Unique continuation and approximate controllability for a degenerate parabolic equation. Appl. Anal. 91 (2012) 140–1425. [Google Scholar]
  15. J.M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example. Asymp. Anal. 44 (2005) 237–257. [Google Scholar]
  16. S. Ervedoza and E. Zuazua, Sharp observability estimates for heat equations. Arch. Ration. Mech. Anal. 202 (2011) 975–1017. [Google Scholar]
  17. H.O. Fattorini and D.L. Russel, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rat. Mech. Anal. 4 (1971) 272–292. [CrossRef] [Google Scholar]
  18. H.O. Fattorini and D.L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quart. Appl. Math. 32 (1974/75) 45–69. [CrossRef] [Google Scholar]
  19. H.O. Fattorini, Boundary control of temperature distributions in a parallelepipedon. SIAM J. Control 13 (1975) 1. [CrossRef] [Google Scholar]
  20. E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5 (2000) 465–514. [Google Scholar]
  21. O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit. J. Funct. Anal. 258 (2010) 852–868. [Google Scholar]
  22. O. Glass and S. Guerrero, Uniform controllability of a transport equation in zero diffusion-dispersion limit. Math. Models Methods Appl. Sci. 19 (2009) 1567–1601. [Google Scholar]
  23. S. Guerrero and G. Lebeau, Singular optimal control for a transport-diffusion equation. Commun. Part. Differ. Equ. 32 (2007) 1813–1836. [CrossRef] [Google Scholar]
  24. M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations. SIAM J. Control Optim. 52 (2014) 2037–2054. [Google Scholar]
  25. E.N. Güichal, A lower bound of the norm of the control operator for the heat equation. J. Math. Anal. Appl. 110 (1985) 519–527. [Google Scholar]
  26. S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems. J. Math. Anal. Appl. 158 (1991) 487–508. [Google Scholar]
  27. A. Haraux, in Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987-1988). Vol. 220 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow (1991) 241–271. [Google Scholar]
  28. E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Band 1: Gewöhnliche Differentialgleichungen, 3rd ed. Chelsea Publishing Company, New York, 1948. [Google Scholar]
  29. V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005. [CrossRef] [Google Scholar]
  30. I. Krasikov, On the Bessel function Jν(x) in the transition zone. LMS J. Comput. Math. 17 (2014) 01. [Google Scholar]
  31. J. Lagnese, Control of wave processes with distributed controls supported on a subregion. SIAM J. Control Optim. 21 (1983) 68–85. [Google Scholar]
  32. L.J. Landau, Bessel functions: monotonicity and bounds. J Lond. Math. Soc. 61 (2000) 197–215. [CrossRef] [Google Scholar]
  33. N.N. Lebedev, Special Functions and their Applications, Dover Publications, New York, 1972. [Google Scholar]
  34. P. Lissy, 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. [Google Scholar]
  35. P. Lissy, 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. [Google Scholar]
  36. L. Lorch and M.E. Muldoon, Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (2008) 221–233. [Google Scholar]
  37. L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time. J. Differ. Equ. 204 (2004) 202–226. [Google Scholar]
  38. L. Miller, How violent are fast controls for Schrödinger and plate vibrations? Arch. Ration. Mech. Anal. 172 (2004) 429–456. [Google Scholar]
  39. L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425–444. [Google Scholar]
  40. L. Miller, The control transmutation method and the cost of fast controls. SIAM J. Control Optim. 45 (2006) 762–772. [Google Scholar]
  41. Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation. J. Fourier Anal. Appl. 19 (2013) 514–544. [Google Scholar]
  42. Y. Privat, E. Trélat and E. Zuazua, Optimal shape and location of sensors for parabolic equations with random initial data. Arch. Ration. Mech. Anal. 216 (2015) 921–981. [Google Scholar]
  43. C.K. Qu and R. Wong, Best possible upper and lower bounds for the zeros of the Bessel function Jν (x). Trans. Am. Math. Soc. 351 (1999) 2833–59. [Google Scholar]
  44. L. Schwartz, Étude des sommes d’exponentielles, deuxième édition. Paris, Hermann (1959). [Google Scholar]
  45. T.I. Seidman, Two results on exact boundary control of parabolic equations. Appl. Math. Optim. 11 (1984) 145–152. [Google Scholar]
  46. T.I. Seidman, S.A. Avdonin and S.A. Ivanov, The window problem for series of complex exponentials. J. Fourier Anal. Appl. 6 (2000) 233–254. [Google Scholar]
  47. T.I. Seidman, How violent are fast controls? Math. Control Signals Syst. 1 (1988) 89–95. [CrossRef] [Google Scholar]
  48. G. Tenenbaum, M. Tucsnak, New blow-up rates for fast controls of Schrodinger and heat equations. J. Differ. Equ. 243 (2007) 70–100. [Google Scholar]
  49. G. Tenenbaum and M. Tucsnak, On the null controllability of diffusion equations, ESAIM: COCV 17 (2011) 1088–1100. [CrossRef] [EDP Sciences] [Google Scholar]
  50. G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge, England (1944). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.