Open Access
Volume 29, 2023
Article Number 15
Number of page(s) 34
Published online 27 February 2023
  1. A. Ankiewicz and N. Akhmediev, Dissipative Solitons: From Optics to Biology and Medicine. Springer, Berlin (2008). [Google Scholar]
  2. P. Baras and J. Goldstein, Remarks on the inverse square potential in quantum mechanics, in: Differential Equations, Birmingham, (1983), in: North-Holland Math. Stud., 92, North-Holland, Amsterdam (1984) 31-35. [CrossRef] [Google Scholar]
  3. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory. Vol. 83 of Math. Sci. Springer, New York (1989). [Google Scholar]
  4. P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications. Mem. Amer. Math. Soc. 239 (2016) 1133. [Google Scholar]
  5. T. Carleman, Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles a deux variables indépendantes. Ark. Mat. Astr. Fys. 26 (1939) 17. [Google Scholar]
  6. M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (1993) 851-1112. [CrossRef] [Google Scholar]
  7. Q. Du, M.D. Gunzburger and J.S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (1992) 54-81. [CrossRef] [MathSciNet] [Google Scholar]
  8. C.L. Epstein and R. Mazzeo, Degenerate diffusion operators arising in population biology. Ann. Math. Stud., Vol. 185, Princeton University Press, Princeton, NJ (2013) [Google Scholar]
  9. S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential. Comm. Partial Differ. Equ. 33 (2008) 1996-2019. [CrossRef] [Google Scholar]
  10. X. Fu, Null controllability for the parabolic equation with a complex principal part. J. Funct. Anal. 257 (2009) 1333-1354. [CrossRef] [MathSciNet] [Google Scholar]
  11. X. Fu and X. Liu, A weighted identity for stochastic partial differential operators and its applications. J. Differ. Equ. 262 (2017) 3551-3582. [CrossRef] [Google Scholar]
  12. X. Fu and X. Liu, Controllability and observability of some stochastic complex Ginzburg-Landau equations. SIAM J. Control Optim. 55 (2017) 1102-1127. [CrossRef] [MathSciNet] [Google Scholar]
  13. X. Fu, Q. Lu and X. Zhang, Carleman estimates for Second Order Partial Differential Operators and Applications. A Unified Approach. Springer, Cham (2019). [Google Scholar]
  14. A.V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes Series 34, Seoul National University, Seoul Korea (1996). [Google Scholar]
  15. Z. Ge, Controllability and observability of stochastic singular systems in Banach spaces. J. Syst. Sci. Complex. 35 (2022) 194-204. [CrossRef] [MathSciNet] [Google Scholar]
  16. V.L. Ginzburg and L.D. Landau, On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20 (1950) 1064-1082 (in Russian); L.D. Landau, Collected Papers, Pergamon Press, Oxford, (1965) 138-167 (in English). [Google Scholar]
  17. P. Hagan and D. Woodward, Equivalent black volatilities. Appl. Math. Finance 6 (1999) 147-159. [CrossRef] [Google Scholar]
  18. E.I. Kaikina, Stochastic Landau-Ginzburg equation with white-noise boundary conditions of Robin type. Nonlinearity 32 (2019) 4967-4995. [CrossRef] [MathSciNet] [Google Scholar]
  19. N.I. Karachalios and N.B. Zographopoulos, On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence. Calc. Var. Partial Differ. Equ. 25 (2006) 361-393. [CrossRef] [Google Scholar]
  20. X. Liu and Y. Yu, Carleman estimates of some stochastic degenerate parabolic equations and application. SIAM J. Control Optim. 57 (2019) 3527-3552. [CrossRef] [MathSciNet] [Google Scholar]
  21. Q. Lu and X. Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations. Probab. Theory Stoch. Model. Springer, Cham (2021). [Google Scholar]
  22. A.C. Newell and J.A. Whitehead, Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38 (1969) 279-303. [CrossRef] [MathSciNet] [Google Scholar]
  23. L. Rosier and B. Zhang, Controllability of the Ginzburg-Landau equation. C.R. Math. Acad. Sci. Paris 346 (2008) 167-172. [CrossRef] [MathSciNet] [Google Scholar]
  24. L. Rosier and B. Zhang, Null controllability of the complex Ginzburg-Landau equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 649-673. [CrossRef] [MathSciNet] [Google Scholar]
  25. M.C. Santos and T.Y. Tanaka, An insensitizing control problem for the Ginzburg-Landau equation. J. Optim. Theory Appl. 183 (2019) 440-470. [CrossRef] [MathSciNet] [Google Scholar]
  26. L.A. Segel, Distant side-walls cause slow amplitude modulations of cellular convection. J. Fluid Mech. 38 (1969) 203-224. [CrossRef] [Google Scholar]
  27. K. Stewartson and J.T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48 (1971) 529-545. [CrossRef] [MathSciNet] [Google Scholar]
  28. S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48 (2009) 2191-2216. [Google Scholar]
  29. J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete Contin. Dyn. Syst. Ser. S 4 (2011) 761-790. [MathSciNet] [Google Scholar]
  30. J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials. J. Funct. Anal. 254 (2017) 1864-1902. [Google Scholar]
  31. B. Wu, Q. Chen and Z. Wang, Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem. Inverse Probl. 36 (2020) 075014. [CrossRef] [Google Scholar]
  32. L. Yan, B. Wu, S. Lu and Y. Wang, Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. ESAIM: COCV 28 (2022) 43. [CrossRef] [EDP Sciences] [Google Scholar]
  33. X. Zhou, A duality analysis on stochastic partial differential equations. J. Funct. Anal. 103 (1992) 275-293. [CrossRef] [MathSciNet] [Google Scholar]

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