September  2015, 35(9): 4415-4437. doi: 10.3934/dcds.2015.35.4415

Stratified discontinuous differential equations and sufficient conditions for robustness

1. 

Project Commands INRIA Saclay & ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762 Palaiseau, France

Received  March 2014 Published  April 2015

This paper is concerned with state-constrained discontinuous ordinary differential equations for which the corresponding vector field has a set of singularities that forms a stratification of the state domain. Existence of solutions and robustness with respect to external perturbations of the righthand term are investigated. Moreover, notions of regularity for stratifications are discussed.
Citation: Cristopher Hermosilla. Stratified discontinuous differential equations and sufficient conditions for robustness. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4415-4437. doi: 10.3934/dcds.2015.35.4415
References:
[1]

Springer-Verlag New York, Inc., 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

Set-Valued and Variational Analysis, 21 (2013), 377-403. doi: 10.1007/s11228-013-0230-y.  Google Scholar

[3]

Springer, 2004.  Google Scholar

[4]

Network and Heterogeneous Media, 2 (2007), 313-331. doi: 10.3934/nhm.2007.2.313.  Google Scholar

[5]

SIAM Journal on Control, 12 (1974), 624-634. doi: 10.1137/0312046.  Google Scholar

[6]

SIAM Journal on Control and Optimization, 14 (1976), 156-162. doi: 10.1137/0314013.  Google Scholar

[7]

Mathematica Slovaca, 28 (1978), 81-100.  Google Scholar

[8]

J. Differential Equations, 38 (1980), 344-360. doi: 10.1016/0022-0396(80)90012-1.  Google Scholar

[9]

in 8th IFAC Symposium on Nonlinear Control Systems, (2010), 1-29. Google Scholar

[10]

Springer, 1998.  Google Scholar

[11]

Springer, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[12]

Mathematical systems theory, 6 (1972), 289-301. doi: 10.1007/BF01740720.  Google Scholar

[13]

J. Differential Equations, 32 (1979), 149-170. doi: 10.1016/0022-0396(79)90056-1.  Google Scholar

[14]

J. Differential Equations, 32 (1979), 171-185, 171-185. Google Scholar

[15]

J. Differential Equations, 258 (2015), 1430-1460. doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[16]

Journal of Economic Theory, 29 (1983), 22-48. doi: 10.1016/0022-0531(83)90121-7.  Google Scholar

[17]

SIAM Journal on Applied Dynamical Systems, 8 (2009), 624-640. doi: 10.1137/08073113X.  Google Scholar

[18]

Mosc. Math. J, 5 (2005), 125-133.  Google Scholar

[19]

ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 291-307. doi: 10.1051/cocv:2002013.  Google Scholar

[20]

Bull. Amer. Math. Soc., 49 (2012), 475-506. doi: 10.1090/S0273-0979-2012-01383-6.  Google Scholar

[21]

SIAM journal on control and optimization, 27 (1989), 53-82. doi: 10.1137/0327005.  Google Scholar

[22]

in Control and Optimization with PDE Constraints, Springer, 164 (2013), 93-116. doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

[23]

Springer, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[24]

SIAM journal on control and optimization, 25 (1987), 1145-1162. doi: 10.1137/0325062.  Google Scholar

[25]

J. Differential Equations, 88 (1990), 15-29. doi: 10.1016/0022-0396(90)90106-Y.  Google Scholar

[26]

Duke Mathematical Journal, 84 (1996), 497-540. doi: 10.1215/S0012-7094-96-08416-1.  Google Scholar

show all references

References:
[1]

Springer-Verlag New York, Inc., 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

Set-Valued and Variational Analysis, 21 (2013), 377-403. doi: 10.1007/s11228-013-0230-y.  Google Scholar

[3]

Springer, 2004.  Google Scholar

[4]

Network and Heterogeneous Media, 2 (2007), 313-331. doi: 10.3934/nhm.2007.2.313.  Google Scholar

[5]

SIAM Journal on Control, 12 (1974), 624-634. doi: 10.1137/0312046.  Google Scholar

[6]

SIAM Journal on Control and Optimization, 14 (1976), 156-162. doi: 10.1137/0314013.  Google Scholar

[7]

Mathematica Slovaca, 28 (1978), 81-100.  Google Scholar

[8]

J. Differential Equations, 38 (1980), 344-360. doi: 10.1016/0022-0396(80)90012-1.  Google Scholar

[9]

in 8th IFAC Symposium on Nonlinear Control Systems, (2010), 1-29. Google Scholar

[10]

Springer, 1998.  Google Scholar

[11]

Springer, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[12]

Mathematical systems theory, 6 (1972), 289-301. doi: 10.1007/BF01740720.  Google Scholar

[13]

J. Differential Equations, 32 (1979), 149-170. doi: 10.1016/0022-0396(79)90056-1.  Google Scholar

[14]

J. Differential Equations, 32 (1979), 171-185, 171-185. Google Scholar

[15]

J. Differential Equations, 258 (2015), 1430-1460. doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[16]

Journal of Economic Theory, 29 (1983), 22-48. doi: 10.1016/0022-0531(83)90121-7.  Google Scholar

[17]

SIAM Journal on Applied Dynamical Systems, 8 (2009), 624-640. doi: 10.1137/08073113X.  Google Scholar

[18]

Mosc. Math. J, 5 (2005), 125-133.  Google Scholar

[19]

ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 291-307. doi: 10.1051/cocv:2002013.  Google Scholar

[20]

Bull. Amer. Math. Soc., 49 (2012), 475-506. doi: 10.1090/S0273-0979-2012-01383-6.  Google Scholar

[21]

SIAM journal on control and optimization, 27 (1989), 53-82. doi: 10.1137/0327005.  Google Scholar

[22]

in Control and Optimization with PDE Constraints, Springer, 164 (2013), 93-116. doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

[23]

Springer, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[24]

SIAM journal on control and optimization, 25 (1987), 1145-1162. doi: 10.1137/0325062.  Google Scholar

[25]

J. Differential Equations, 88 (1990), 15-29. doi: 10.1016/0022-0396(90)90106-Y.  Google Scholar

[26]

Duke Mathematical Journal, 84 (1996), 497-540. doi: 10.1215/S0012-7094-96-08416-1.  Google Scholar

[1]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012

[2]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021063

[3]

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

[4]

Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047

[5]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001

[6]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[7]

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

[8]

Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021061

[9]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[10]

Yuta Ishii, Kazuhiro Kurata. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021035

[11]

Yu-Hsien Liao. Solutions and characterizations under multicriteria management systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021041

[12]

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

[13]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[14]

Miroslav Bulíček, Victoria Patel, Endre Süli, Yasemin Şengül. Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021053

[15]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071

[16]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313

[17]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[18]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[19]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012

[20]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]