EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 30 (2024) ESAIM: COCV, 30 (2024) 41 References
Open Access
Issue
ESAIM: COCV
Volume 30, 2024
Article Number 41
Number of page(s) 45
DOI https://doi.org/10.1051/cocv/2024030
Published online 08 May 2024
  1. S.P. Hastings, Some mathematical problems from neurobiology. Amer. Math. Monthly 82 (1975) 881–895. [CrossRef] [MathSciNet] [Google Scholar]
  2. J.M. Rogers and A.D. McCulloch, A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41 (1994) 743–757. [CrossRef] [Google Scholar]
  3. T. Breiten and K. Kunisch, Riccati-based feedback control of the monodomain equations with the FitzHugh–Nagumo model. SIAM J. Control Optim. 52 (2014) 4057–4081. [CrossRef] [MathSciNet] [Google Scholar]
  4. T. Breiten and K. Kunisch, Boundary feedback stabilization of the monodomain equations. Math. Control Relat. Fields 7 (2017) 369–391. [CrossRef] [MathSciNet] [Google Scholar]
  5. T. Breiten, K. Kunisch and S.S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh–Nagumo type. SIAM J. Control Optim. 55 (2017) 2684–2713. [CrossRef] [MathSciNet] [Google Scholar]
  6. N.A. Trayanova and L.J. Rantner, New insights into defibrillation of the heart from realistic simulation studies. Eur. Soc. Cardiol. 16 (2014). [Google Scholar]
  7. T. Breiten and K. Kunisch, Compensator design for the monodomain equations with the FitzHugh–Nagumo model. ESAIM Control Optim. Calc. Var. 23 (2017) 241–262. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  8. S. Chowdhury, M. Biswas and R. Dutta, Approximate controllability of the FitzHugh–Nagumo equation in one dimension. J. Differ. Equ. 268 (2020) 3497–3563. [CrossRef] [Google Scholar]
  9. S. Micu, On the controllability of the linearized Benjamin–Bona-Mahony equation. SIAM J. Control Optim. 39 (2001) 1677–1696. [CrossRef] [MathSciNet] [Google Scholar]
  10. L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping. Int. J. Tomogr. Stat. 5 (2007) 79–84. [MathSciNet] [Google Scholar]
  11. S. Chowdhury, M. Ramaswamy and J.-P. Raymond, Controllability and stabilizability of the linearized compressible Navier–Stokes system in one dimension. SIAM J. Control Optim. 50 (2012) 2959–2987. [CrossRef] [MathSciNet] [Google Scholar]
  12. M. Krstic and A. Smyshlyaev, Boundary Control of PDEs, Vol. 16 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008). [Google Scholar]
  13. W. Liu, Boundary feedback stabilization of an unstable heat equation. SIAM J. Control Optim. 42 (2003) 1033–1043. [CrossRef] [MathSciNet] [Google Scholar]
  14. W. Liu, Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation, Vol. 66 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2010). [Google Scholar]
  15. E. Cerpa and J.-M. Coron, Rapid stabilization for a Korteweg–de Vries equation from the left Dirichlet boundary condition. IEEE Trans. Automat. Control 58 (2013) 1688–1695. [CrossRef] [MathSciNet] [Google Scholar]
  16. X. Yu, C. Xu and J. Chu, Local exponential stabilization of Fisher’s equation using the backstepping technique. Syst. Control Lett. 74 (2014) 1–7. [CrossRef] [Google Scholar]
  17. O.M. Aamo, A. Smyshlyaev and M. Krstić, Boundary control of the linearized Ginzburg–Landau model of vortex shedding. SIAM J. Control Optim. 43 (1971) 1953–1971. [Google Scholar]
  18. A. Smyshlyaev and M. Krstic, Backstepping observers for a class of parabolic PDEs. Syst. Control Lett. 54 (2005) 613–625. [CrossRef] [Google Scholar]
  19. R. Vazquez and M. Krstic, Boundary control of coupled reaction-diffusion system with spatially-varying reaction. IFAC- PapersOnLine 49 (2016) 222–227. [CrossRef] [MathSciNet] [Google Scholar]
  20. R. Vazquez and M. Krstic, Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients. IEEE Trans. Automat. Control 62 (2017) 2026–2033. [CrossRef] [MathSciNet] [Google Scholar]
  21. F. Di Meglio, R. Vazquez and M. Krstic, Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Trans. Automat. Control 58 (2013) 3097–3111. [Google Scholar]
  22. L. Hu, F. Di Meglio, R. Vazquez and M. Krstic, Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs. IEEE Trans. Automat. Control 61 (2016) 3301–3314. [Google Scholar]
  23. L. Hu, R. Vazquez, F. Di Meglio and M. Krstic, Boundary exponential stabilization of 1-dimensional inhomogeneous quasi-linear hyperbolic systems. SIAM J. Control Optim. 57 (2019) 963–998. [CrossRef] [MathSciNet] [Google Scholar]
  24. A. Smyshlyaev, E. Cerpa and M. Krstic, Boundary stabilization of a 1-D wave equation with in-domain antidamping. SIAM J. Control Optim. 48 (2010) 4014–4031. [CrossRef] [MathSciNet] [Google Scholar]
  25. R.A. Capistrano-Filho and F.A. Gallego, Asymptotic behavior of Boussinesq system of KdV–KdV type. J. Differ. Equ. 265 (2018) 2341–2374. [CrossRef] [Google Scholar]
  26. J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg–de Vries equation with a Neumann boundary control on the right. J. Math. Pures Appl. 102 (2014) 1080–1120. [CrossRef] [MathSciNet] [Google Scholar]
  27. J.-M. Coron and Q. Lu, Fredholm transform and local rapid stabilization for a Kuramoto-Sivashinsky equation. J. Differ. Equ. 259 (2015) 3683–3729. [CrossRef] [Google Scholar]
  28. J.-M. Coron, R. Vazquez, M. Krstic and G. Bastin, Local exponential H2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping. SIAM J. Control Optim. 51 (2013) 2005–2035. [CrossRef] [MathSciNet] [Google Scholar]
  29. J.-M. Coron and S. Xiang, Small-time global stabilization of the viscous Burgers equation with three scalar controls. J. Math. Pures Appl. 151 (2021) 212–256. [CrossRef] [MathSciNet] [Google Scholar]
  30. J.-M. Coron, L. Hu and G. Olive, Stabilization and controllability of first-order integro-differential hyperbolic equations. J. Funct. Anal. 271 (2016) 3554–3587. [CrossRef] [MathSciNet] [Google Scholar]
  31. J.M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators. SIAM J. Control Optim. 43 (2005) 2233–2244. [CrossRef] [MathSciNet] [Google Scholar]
  32. E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg–de Vries equation. Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 655–668. [MathSciNet] [Google Scholar]
  33. A. Smyshlyaev and M. Krstic, On control design for PDEs with space-dependent diffusivity or time-dependent reactivity. Automatica J. IFAC 41 (2005) 1601–1608. [CrossRef] [MathSciNet] [Google Scholar]
  34. J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach. Arch. Ration. Mech. Anal. 225 (2017) 993–1023. [CrossRef] [MathSciNet] [Google Scholar]
  35. S. Xiang, Small-time local stabilization for a Korteweg–de Vries equation. Syst. Control Lett. 111 (2018) 64–69. [CrossRef] [Google Scholar]
  36. S. Xiang, Null controllability of a linearized Korteweg–de Vries equation by backstepping approach. SIAM J. Control Optim. 57 (2019) 1493–1515. [CrossRef] [MathSciNet] [Google Scholar]
  37. A.J.V. Brandão, E. Fernández-Cara, P.M.D. Magalhães and M.A. Rojas-Medar, Theoretical analysis and control results for the FitzHugh–Nagumo equation. Electron. J. Differ. Equ. 20 (2008) 164. [Google Scholar]
  38. E. Casas, C. Ryll and F. Triöltzsch, Second order and stability analysis for optimal sparse control of the FitzHugh–Nagumo equation. SIAM J. Control Optim. 53 (2015) 2168–2202. [CrossRef] [MathSciNet] [Google Scholar]
  39. N. Chamakuri, K. Kunisch and G. Plank, Optimal control approach to termination of re-entry waves in cardiac electrophysiology. J. Math. Biol. 67 (2013) 359–388. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  40. N. Chamakuri, K. Kunisch and G. Plank, Application of optimal control to the cardiac defibrillation problem using a physiological model of cellular dynamics. Appl. Numer. Math. 95 (2015) 130–139. [CrossRef] [MathSciNet] [Google Scholar]
  41. K. Kunisch, K. Pieper and A. Rund, Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach. ESAIM Math. Model. Numer. Anal. 50 (2016) 381–414. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  42. K. Kunisch and A. Rund, Time optimal control of the monodomain model in cardiac electrophysiology. IMA J. Appl. Math. 80 (2015) 1664–1683. [CrossRef] [MathSciNet] [Google Scholar]
  43. K. Kunisch and M. Wagner, Optimal control of the bidomain system (I): the monodomain approximation with the Rogers–McCulloch model. Nonlinear Anal. Real World Appl. 13 (2012) 1525–1550. [CrossRef] [MathSciNet] [Google Scholar]
  44. K. Kunisch and L. Wang, Time optimal controls of the linear Fitzhugh–Nagumo equation with pointwise control constraints. J. Math. Anal. Appl. 395 (2012) 114–130. [CrossRef] [MathSciNet] [Google Scholar]
  45. T. Berger, T. Breiten, M. Puche and T. Reis, Funnel control for the monodomain equations with the FitzHugh–Nagumo model. J. Differ. Equ. 286 (2021) 164–214. [CrossRef] [Google Scholar]
  46. K. Kunisch and D.A. Souza, On the one-dimensional nonlinear monodomain equations with moving controls. J. Math. Pures Appl. 117 (2018) 94–122. [CrossRef] [MathSciNet] [Google Scholar]
  47. S. Guerrero and O.Y. Imanuvilov, Remarks on noncontrollability of the heat equation with memory. ESAIM Control Optim. Calc. Var. 19 (2013) 288–300. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  48. H. Arfaoui, F. Ben Belgacem, H. El Fekih and J.-P. Raymond, Boundary stabilizability of the linearized viscous Saint–Venant system. Discrete Contin. Dyn. Syst. Ser. B 15 (2011) 491–511. [MathSciNet] [Google Scholar]
  49. E. Fernández-Cara, J. Lucas F. Machado and D.A. Souza, Non null controllability of Stokes equations with memory. ESAIM Control Optim. Calc. Var. 26 (2020) Paper No. 72, 18. [Google Scholar]
  50. A. Doubova and E. Fernández-Cara, On the control of viscoelastic Jeffreys fluids. Syst. Control Lett. 61 (2012) 573–579. [CrossRef] [Google Scholar]
  51. D. Maity, D. Mitra and M. Renardy, Lack of null controllability of viscoelastic flows. ESAIM Control Optim. Calc. Var. 25 (2019) Paper No. 60, 26. [CrossRef] [EDP Sciences] [Google Scholar]
  52. S. Chowdhury, R. Dutta and S. Majumdar, Boundary stabilizability of the linearized compressible Navier–Stokes system in one dimension by backstepping approach. SIAM J. Control Optim. 59 (2021) 2147–2173. [CrossRef] [MathSciNet] [Google Scholar]
  53. M. Renardy, A note on a class of observability problems for PDEs. Syst. Control Lett. 58 (2009) 183–187. [CrossRef] [Google Scholar]
  54. L. Hu and Z. Wang, On boundary control of a hyperbolic system with a vanishing characteristic speed. ESAIM Control Optim. Calc. Var. 22 (2016) 134–147. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  55. J. Keener and J. Sneyd, Mathematical Physiology. Vol. II: Systems Physiology, Vol. 8 of Interdisciplinary Applied Mathematics, 2nd edn. Springer, New York (2009). [CrossRef] [Google Scholar]
  56. F. Schlögl and E. Schöll, A relation between cumulants of a thermodynamic variable and the state equation. Z. Phys. B 51 (1983) 61–64. [CrossRef] [MathSciNet] [Google Scholar]
  57. A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edn. Systems & Control: Foundations & Applications. Birkhauser Boston, Inc., Boston, MA (2007). [CrossRef] [Google Scholar]
  58. S. Kesavan and J.-P. Raymond, On a degenerate Riccati equation. Control Cybernet. 38 (2009) 1393–1410. [MathSciNet] [Google Scholar]
  59. J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations. SIAM J. Control Optim. 43 (2004) 549–569. [Google Scholar]
  60. J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations. Commun. Contemp. Math. 8 (2006) 535–567. [CrossRef] [Google Scholar]
  61. M. Schmidt and E. Trélat, Controllability of Couette flows. Commun. Pure Appl. Anal. 5 (2006) 201–211. [CrossRef] [MathSciNet] [Google Scholar]
  62. D. Tsubakino, M. Krstic and S. Hara, Backstepping control for parabolic pdes with in-domain actuation, in 2012 American Control Conference (ACC) (2012) 2226–2231. [CrossRef] [Google Scholar]
  63. J.-M. Coron, L. Gagnon and M. Morancey, Rapid stabilization of a linearized bilinear 1-D Schrödinger equation. J. Math. Pures Appl. 115 (2018) 24–73. [CrossRef] [MathSciNet] [Google Scholar]
  64. C. Zhang, Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. working paper or preprint (2018). [Google Scholar]
  65. F.W. Chaves-Silva, X. Zhang and E. Zuazua, Controllability of evolution equations with memory. SIAM J. Control Optim. 55 (2017) 2437–2459. [CrossRef] [MathSciNet] [Google Scholar]
  66. F.W. Chaves-Silva, L. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control. J. Math. Pures Appl. 101 (2014) 198–222. [Google Scholar]
  67. L. Jadachowski, T. Meurer and A. Kugi, Backstepping observers for linear PDEs on higher-dimensional spatial domains. Automatica J. IFAC 51 (2015) 85–97. [CrossRef] [MathSciNet] [Google Scholar]
  68. R. Vazquez and M. Krstic, Explicit integral operator feedback for local stabilization of nonlinear thermal convection loop PDEs. Syst. Control Lett. 55 (2006) 624–632. [CrossRef] [Google Scholar]
  69. R. Vázquez, E. Trélat and J.-M. Coron, Control for fast and stable laminar-to-high-Reynolds-numbers transfer in a 2D Navier–Stokes channel flow. Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 925–956. [MathSciNet] [Google Scholar]
  70. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Vol. 68 of Applied Mathematical Sciences, 2nd edn. Springer-Verlag, New York (1997). [CrossRef] [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.