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Strict Lyapunov functions for semilinear parabolic partial differential equations
1. | Team INRIA DISCO, CNRS-Supelec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France |
2. | Department of Automatic Control, Gipsa-lab, 961 rue de la Houille Blanche, BP 46, 38402 Grenoble Cedex |
References:
[1] |
J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Applied Mathematical Sciences, 83 (1989).
|
[2] |
T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998).
|
[3] |
X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, Journal of Differential Equations, 78 (1989), 160.
|
[4] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,, SIAM Journal on Control and Optimization, 47 (2008), 1460.
doi: 10.1137/070706847. |
[5] |
J.-M. Coron and B. d'Andréa-Novel, Stabilization of a rotating body beam without damping,, IEEE Transactions on Automatic Control, 43 (1998), 608.
doi: 10.1109/9.668828. |
[6] |
J.-M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Transactions on Automatic Control, 52 (2007), 2.
doi: 10.1109/TAC.2006.887903. |
[7] |
J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM Journal on Control and Optimization, 43 (2004), 549.
doi: 10.1137/S036301290342471X. |
[8] |
J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations,, Commun. Contemp. Math., 8 (2006), 535.
doi: 10.1142/S0219199706002209. |
[9] |
A. K. Dramé, D. Dochain and J. J. Winkin, Asymptotic behavior and stability for solutions of a biochemical reactor distributed parameter model,, IEEE Transactions on Automatic Control, 53 (2008), 412.
doi: 10.1109/TAC.2007.914948. |
[10] |
O. V. Iftime and M. A. Demetriou, Optimal control of switched distributed parameter systems with spatially scheduled actuators,, Automatica J. IFAC, 45 (2009), 312.
doi: 10.1016/j.automatica.2008.07.012. |
[11] |
I. Karafyllis, P. Pepe and Z.-P. Jiang, Input-to-output stability for systems described by retarded functional differential equations,, European Journal of Control, 14 (2008), 539.
doi: 10.3166/ejc.14.539-555. |
[12] |
M. Krstic and A. Smyshlyaev, "Boundary Control of PDEs. A Course on Backstepping Designs,", Advances in Design and Control, 16 (2008).
|
[13] |
M. Krstic and A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs. I. Lyapunov design,, IEEE Transactions on Automatic Control, 53 (2008), 1575.
doi: 10.1109/TAC.2008.927798. |
[14] |
Z.-H. Luo, B.-Z. Guo and O. Morgul, "Stability and Stabilization of Infinite Dimensional Systems with Applications,", Communications and Control Engineering Series, (1999).
|
[15] |
M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions,", Communications and Control Engineering Series, (2009).
|
[16] |
D. Matignon and C. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems,, ESAIM Control Optim. Cal. Var., 11 (2005), 487.
doi: 10.1051/cocv:2005016. |
[17] |
F. Mazenc, M. Malisoff and O. Bernard, A simplified design for strict Lyapunov functions under Matrosov conditions,, IEEE Transactions on Automatic Control, 54 (2009), 177.
doi: 10.1109/TAC.2008.2008353. |
[18] |
F. Mazenc, M. Malisoff and Z. Lin, Further results on input-to-state stability for nonlinear systems with delayed feedbacks,, Automatica J. IFAC, 44 (2008), 2415.
doi: 10.1016/j.automatica.2008.01.024. |
[19] |
F. Mazenc and D. Nesic, Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem,, Mathematics of Control, 19 (2007), 151.
doi: 10.1007/s00498-007-0015-7. |
[20] |
F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type ut$=\delta u+$ |$u$|p-1 $u$,, Duke Math. J., 86 (1997), 143.
doi: 10.1215/S0012-7094-97-08605-1. |
[21] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
|
[22] |
P. Pepe, Input-to-state stabilization of stabilizable, time-delay, control-affine, nonlinear systems,, IEEE Transactions on Automatic Control, 54 (2009), 1688.
doi: 10.1109/TAC.2009.2020642. |
[23] |
P. Pepe and H. Ito, On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign,, in, (2010), 190. |
[24] |
C. Prieur and F. Mazenc, ISS Lyapunov functions for time-varying hyperbolic partial differential equations,, submitted for publication, (2011). |
[25] |
M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM Journal on Control, 12 (1974), 500.
doi: 10.1137/0312038. |
[26] |
A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. II. Estimation-based designs,, Automatica J. IFAC, 43 (2007), 1543.
doi: 10.1016/j.automatica.2007.02.014. |
[27] |
A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. III. Output feedback examples with swapping identifiers,, Automatica J. IFAC, 43 (2007), 1557.
doi: 10.1016/j.automatica.2007.02.015. |
[28] |
E. D. Sontag, Input to state stability: Basic concepts and results,, Nonlinear and Optimal Control Theory, (2007), 163. |
[29] |
M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117,, Springer-Verlag, (1997).
|
show all references
References:
[1] |
J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Applied Mathematical Sciences, 83 (1989).
|
[2] |
T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998).
|
[3] |
X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, Journal of Differential Equations, 78 (1989), 160.
|
[4] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,, SIAM Journal on Control and Optimization, 47 (2008), 1460.
doi: 10.1137/070706847. |
[5] |
J.-M. Coron and B. d'Andréa-Novel, Stabilization of a rotating body beam without damping,, IEEE Transactions on Automatic Control, 43 (1998), 608.
doi: 10.1109/9.668828. |
[6] |
J.-M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Transactions on Automatic Control, 52 (2007), 2.
doi: 10.1109/TAC.2006.887903. |
[7] |
J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM Journal on Control and Optimization, 43 (2004), 549.
doi: 10.1137/S036301290342471X. |
[8] |
J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations,, Commun. Contemp. Math., 8 (2006), 535.
doi: 10.1142/S0219199706002209. |
[9] |
A. K. Dramé, D. Dochain and J. J. Winkin, Asymptotic behavior and stability for solutions of a biochemical reactor distributed parameter model,, IEEE Transactions on Automatic Control, 53 (2008), 412.
doi: 10.1109/TAC.2007.914948. |
[10] |
O. V. Iftime and M. A. Demetriou, Optimal control of switched distributed parameter systems with spatially scheduled actuators,, Automatica J. IFAC, 45 (2009), 312.
doi: 10.1016/j.automatica.2008.07.012. |
[11] |
I. Karafyllis, P. Pepe and Z.-P. Jiang, Input-to-output stability for systems described by retarded functional differential equations,, European Journal of Control, 14 (2008), 539.
doi: 10.3166/ejc.14.539-555. |
[12] |
M. Krstic and A. Smyshlyaev, "Boundary Control of PDEs. A Course on Backstepping Designs,", Advances in Design and Control, 16 (2008).
|
[13] |
M. Krstic and A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs. I. Lyapunov design,, IEEE Transactions on Automatic Control, 53 (2008), 1575.
doi: 10.1109/TAC.2008.927798. |
[14] |
Z.-H. Luo, B.-Z. Guo and O. Morgul, "Stability and Stabilization of Infinite Dimensional Systems with Applications,", Communications and Control Engineering Series, (1999).
|
[15] |
M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions,", Communications and Control Engineering Series, (2009).
|
[16] |
D. Matignon and C. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems,, ESAIM Control Optim. Cal. Var., 11 (2005), 487.
doi: 10.1051/cocv:2005016. |
[17] |
F. Mazenc, M. Malisoff and O. Bernard, A simplified design for strict Lyapunov functions under Matrosov conditions,, IEEE Transactions on Automatic Control, 54 (2009), 177.
doi: 10.1109/TAC.2008.2008353. |
[18] |
F. Mazenc, M. Malisoff and Z. Lin, Further results on input-to-state stability for nonlinear systems with delayed feedbacks,, Automatica J. IFAC, 44 (2008), 2415.
doi: 10.1016/j.automatica.2008.01.024. |
[19] |
F. Mazenc and D. Nesic, Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem,, Mathematics of Control, 19 (2007), 151.
doi: 10.1007/s00498-007-0015-7. |
[20] |
F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type ut$=\delta u+$ |$u$|p-1 $u$,, Duke Math. J., 86 (1997), 143.
doi: 10.1215/S0012-7094-97-08605-1. |
[21] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
|
[22] |
P. Pepe, Input-to-state stabilization of stabilizable, time-delay, control-affine, nonlinear systems,, IEEE Transactions on Automatic Control, 54 (2009), 1688.
doi: 10.1109/TAC.2009.2020642. |
[23] |
P. Pepe and H. Ito, On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign,, in, (2010), 190. |
[24] |
C. Prieur and F. Mazenc, ISS Lyapunov functions for time-varying hyperbolic partial differential equations,, submitted for publication, (2011). |
[25] |
M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM Journal on Control, 12 (1974), 500.
doi: 10.1137/0312038. |
[26] |
A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. II. Estimation-based designs,, Automatica J. IFAC, 43 (2007), 1543.
doi: 10.1016/j.automatica.2007.02.014. |
[27] |
A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. III. Output feedback examples with swapping identifiers,, Automatica J. IFAC, 43 (2007), 1557.
doi: 10.1016/j.automatica.2007.02.015. |
[28] |
E. D. Sontag, Input to state stability: Basic concepts and results,, Nonlinear and Optimal Control Theory, (2007), 163. |
[29] |
M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117,, Springer-Verlag, (1997).
|
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