-
Previous Article
Complete controllability for a class of fractional evolution equations with uncertainty
- EECT Home
- This Issue
-
Next Article
Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations
Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation
Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan |
$ \begin{equation*} \begin{cases} \dfrac{du}{dt}(t)+Au(t) = F(u(t), (u(t-\tau_n))_{n\in\mathbb{N}}), & t>0, \\ u(t) = u_0(t), & t \in \bigcup\limits_{n \in \mathbb{N}}[-\tau_n,0], \end{cases} \end{equation*} $ |
$ X $ |
$ -A $ |
$ C_0 $ |
$ F: X \times X^\mathbb{N} \to X $ |
References:
[1] |
F. Alabau-Boussouira, S. Nicaise and C. Pignotti, Exponential stability of the wave equation with memory and time delay, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 1–22.
doi: 10.1007/978-3-319-11406-4_1. |
[2] |
K. Ammari, S. Nicaise and C. Pignotti,
Feedback boundary stabilization of wave equations with interior delay, Systems Control Lett., 59 (2010), 623-628.
doi: 10.1016/j.sysconle.2010.07.007. |
[3] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[4] |
A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, AK Peters, Ltd., Wellesley, MA, 2005. |
[5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[6] |
R. Datko,
Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.
doi: 10.1137/0326040. |
[7] |
R. Datko, J. Lagnese and M. P. Polis,
An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.
doi: 10.1137/0324007. |
[8] |
H. I. Freedman and X.-Q. Zhao,
Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.
doi: 10.1006/jdeq.1997.3264. |
[9] |
G. Friesecke,
Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Differential Equations, 5 (1993), 89-103.
doi: 10.1007/BF01063736. |
[10] |
A. Guesmia,
Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.
doi: 10.1093/imamci/dns039. |
[11] |
A. Inoue, T. Miyakawa and K. Yoshida,
Some properties of solutions for semilinear heat equations with time lag, J. Differential Equations, 24 (1977), 383-396.
doi: 10.1016/0022-0396(77)90007-9. |
[12] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, 1994. |
[13] |
D. Li and S. Guo,
Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect, J. Math. Anal. Appl., 467 (2018), 1080-1099.
doi: 10.1016/j.jmaa.2018.07.050. |
[14] |
D. Li and S. Guo,
Traveling wavefronts in a reaction-diffusion model with chemotaxis and nonlocal delay effect, Nonlinear Anal. Real World Appl., 45 (2019), 736-754.
doi: 10.1016/j.nonrwa.2018.08.001. |
[15] |
J. H. Lightbourne and S. M. Rankin,
Global existence for a delay differential equation, J. Differential Equations, 40 (1981), 186-192.
doi: 10.1016/0022-0396(81)90017-6. |
[16] |
W. Liu,
Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56.
doi: 10.3934/dcdsb.2002.2.47. |
[17] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[18] |
S. Nicaise and C. Pignotti,
Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.
|
[19] |
S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 2011, No. 41, 20 pp. |
[20] |
S. Nicaise and C. Pignotti,
Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.
doi: 10.1007/s00498-014-0130-1. |
[21] |
S. Nicaise and C. Pignotti,
Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15 (2015), 107-129.
doi: 10.1007/s00028-014-0251-5. |
[22] |
S. Nicaise and C. Pignotti,
Stability of the wave equation with localized Kelvi–-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.
doi: 10.3934/dcdss.2016029. |
[23] |
S. Nicaise and C. Pignotti,
Well-posedness and stability results for nonlinear abstract evolution equations with time delays, J. Evol. Equ., 18 (2018), 947-971.
doi: 10.1007/s00028-018-0427-5. |
[24] |
S. M. Oliva,
Reaction-diffusion equations with nonlinear boundary delay, J. Dynam. Differential Equations, 11 (1999), 279-296.
doi: 10.1023/A:1021929413376. |
[25] |
C. V. Pao,
Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.
doi: 10.1006/jmaa.1996.0111. |
[26] |
C. V. Pao,
Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104.
doi: 10.1016/S1468-1218(03)00018-X. |
[27] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[28] |
C. Pignotti,
A note on stabilization of locally damped wave equations with time delay, Systems Control Lett., 61 (2012), 92-97.
doi: 10.1016/j.sysconle.2011.09.016. |
[29] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007. |
[30] |
S. Ruan and X.-Q. Zhao,
Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations, 156 (1999), 71-92.
doi: 10.1006/jdeq.1998.3599. |
[31] |
B. Said-Houari and A. Soufyane,
Stability result of the Timoshenko system with delay and boundary feedback, IMA J. Math. Control Inform., 29 (2012), 383-398.
doi: 10.1093/imamci/dnr043. |
[32] |
G. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[33] |
K. Yoshii, Solvability in abstract evolution equations with countable time delays in Banach spaces: Lobal Lipschitz perturbation, preprint, 2020. Google Scholar |
[34] |
E. Zuazua,
Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.
doi: 10.1080/03605309908820684. |
show all references
References:
[1] |
F. Alabau-Boussouira, S. Nicaise and C. Pignotti, Exponential stability of the wave equation with memory and time delay, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 1–22.
doi: 10.1007/978-3-319-11406-4_1. |
[2] |
K. Ammari, S. Nicaise and C. Pignotti,
Feedback boundary stabilization of wave equations with interior delay, Systems Control Lett., 59 (2010), 623-628.
doi: 10.1016/j.sysconle.2010.07.007. |
[3] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[4] |
A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, AK Peters, Ltd., Wellesley, MA, 2005. |
[5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[6] |
R. Datko,
Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.
doi: 10.1137/0326040. |
[7] |
R. Datko, J. Lagnese and M. P. Polis,
An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.
doi: 10.1137/0324007. |
[8] |
H. I. Freedman and X.-Q. Zhao,
Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.
doi: 10.1006/jdeq.1997.3264. |
[9] |
G. Friesecke,
Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Differential Equations, 5 (1993), 89-103.
doi: 10.1007/BF01063736. |
[10] |
A. Guesmia,
Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.
doi: 10.1093/imamci/dns039. |
[11] |
A. Inoue, T. Miyakawa and K. Yoshida,
Some properties of solutions for semilinear heat equations with time lag, J. Differential Equations, 24 (1977), 383-396.
doi: 10.1016/0022-0396(77)90007-9. |
[12] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, 1994. |
[13] |
D. Li and S. Guo,
Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect, J. Math. Anal. Appl., 467 (2018), 1080-1099.
doi: 10.1016/j.jmaa.2018.07.050. |
[14] |
D. Li and S. Guo,
Traveling wavefronts in a reaction-diffusion model with chemotaxis and nonlocal delay effect, Nonlinear Anal. Real World Appl., 45 (2019), 736-754.
doi: 10.1016/j.nonrwa.2018.08.001. |
[15] |
J. H. Lightbourne and S. M. Rankin,
Global existence for a delay differential equation, J. Differential Equations, 40 (1981), 186-192.
doi: 10.1016/0022-0396(81)90017-6. |
[16] |
W. Liu,
Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56.
doi: 10.3934/dcdsb.2002.2.47. |
[17] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[18] |
S. Nicaise and C. Pignotti,
Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.
|
[19] |
S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 2011, No. 41, 20 pp. |
[20] |
S. Nicaise and C. Pignotti,
Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.
doi: 10.1007/s00498-014-0130-1. |
[21] |
S. Nicaise and C. Pignotti,
Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15 (2015), 107-129.
doi: 10.1007/s00028-014-0251-5. |
[22] |
S. Nicaise and C. Pignotti,
Stability of the wave equation with localized Kelvi–-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.
doi: 10.3934/dcdss.2016029. |
[23] |
S. Nicaise and C. Pignotti,
Well-posedness and stability results for nonlinear abstract evolution equations with time delays, J. Evol. Equ., 18 (2018), 947-971.
doi: 10.1007/s00028-018-0427-5. |
[24] |
S. M. Oliva,
Reaction-diffusion equations with nonlinear boundary delay, J. Dynam. Differential Equations, 11 (1999), 279-296.
doi: 10.1023/A:1021929413376. |
[25] |
C. V. Pao,
Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.
doi: 10.1006/jmaa.1996.0111. |
[26] |
C. V. Pao,
Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104.
doi: 10.1016/S1468-1218(03)00018-X. |
[27] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[28] |
C. Pignotti,
A note on stabilization of locally damped wave equations with time delay, Systems Control Lett., 61 (2012), 92-97.
doi: 10.1016/j.sysconle.2011.09.016. |
[29] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007. |
[30] |
S. Ruan and X.-Q. Zhao,
Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations, 156 (1999), 71-92.
doi: 10.1006/jdeq.1998.3599. |
[31] |
B. Said-Houari and A. Soufyane,
Stability result of the Timoshenko system with delay and boundary feedback, IMA J. Math. Control Inform., 29 (2012), 383-398.
doi: 10.1093/imamci/dnr043. |
[32] |
G. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[33] |
K. Yoshii, Solvability in abstract evolution equations with countable time delays in Banach spaces: Lobal Lipschitz perturbation, preprint, 2020. Google Scholar |
[34] |
E. Zuazua,
Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.
doi: 10.1080/03605309908820684. |
[1] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
[2] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[3] |
Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191 |
[4] |
Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 |
[5] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
[6] |
John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026 |
[7] |
Akio Matsumot, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021069 |
[8] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[9] |
Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 |
[10] |
Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973 |
[11] |
Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005 |
[12] |
Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 |
[13] |
Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513 |
[14] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[15] |
Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075 |
[16] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[17] |
Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493 |
[18] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[19] |
Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 |
[20] |
Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195 |
2019 Impact Factor: 0.953
Tools
Article outline
[Back to Top]