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 - A, 2015, 35 (9) : 4415-4437. doi: 10.3934/dcds.2015.35.4415
References:
[1]

J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory,, Springer-Verlag New York, (1984).  doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

R. C. Barnard and P. R. Wolenski, Flow invariance on stratified domains,, Set-Valued and Variational Analysis, 21 (2013), 377.  doi: 10.1007/s11228-013-0230-y.  Google Scholar

[3]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, vol. 43,, Springer, (2004).   Google Scholar

[4]

A. Bressan and Y. Hong, Optimal control problems on stratified domains,, Network and Heterogeneous Media, 2 (2007), 313.  doi: 10.3934/nhm.2007.2.313.  Google Scholar

[5]

P. Brunovskỳ, The closed-loop time-optimal control. I: Optimality,, SIAM Journal on Control, 12 (1974), 624.  doi: 10.1137/0312046.  Google Scholar

[6]

P. Brunovskỳ, The closed-loop time optimal control. II: Stability,, SIAM Journal on Control and Optimization, 14 (1976), 156.  doi: 10.1137/0314013.  Google Scholar

[7]

P. Brunovskỳ, Every normal linear system has a regular time-optimal synthesis,, Mathematica Slovaca, 28 (1978), 81.   Google Scholar

[8]

P. Brunovskỳ, Regular synthesis for the linear-quadratic optimal control problem with linear control constraints,, J. Differential Equations, 38 (1980), 344.  doi: 10.1016/0022-0396(80)90012-1.  Google Scholar

[9]

F. Clarke, Discontinuous feedback and nonlinear systems,, in 8th IFAC Symposium on Nonlinear Control Systems, (2010), 1.   Google Scholar

[10]

F. Clarke, Y. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178,, Springer, (1998).   Google Scholar

[11]

A. F. Filippov and F. M. Arscott, Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18,, Springer, (1988).  doi: 10.1007/978-94-015-7793-9.  Google Scholar

[12]

O. Hájek, Terminal manifolds and switching locus,, Mathematical systems theory, 6 (1972), 289.  doi: 10.1007/BF01740720.  Google Scholar

[13]

O. Hájek, Discontinuous differential equations, I,, J. Differential Equations, 32 (1979), 149.  doi: 10.1016/0022-0396(79)90056-1.  Google Scholar

[14]

O. Hájek, Discontinuous differential equations, II,, J. Differential Equations, 32 (1979), 171.   Google Scholar

[15]

C. Hermosilla and H. Zidani, Infinite horizon problem on stratifiable state-constraints set,, J. Differential Equations, 258 (2015), 1430.  doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[16]

S. Honkapohja and T. Ito, Stability with regime switching,, Journal of Economic Theory, 29 (1983), 22.  doi: 10.1016/0022-0531(83)90121-7.  Google Scholar

[17]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields,, SIAM Journal on Applied Dynamical Systems, 8 (2009), 624.  doi: 10.1137/08073113X.  Google Scholar

[18]

V. Y. Kaloshin, A geometric proof of the existence of whitney stratifications,, Mosc. Math. J, 5 (2005), 125.   Google Scholar

[19]

A. Marigo and B. Piccoli, Regular syntheses and solutions to discontinuous odes,, ESAIM: Control, 7 (2002), 291.  doi: 10.1051/cocv:2002013.  Google Scholar

[20]

J. Mather, Notes on topological stability,, Bull. Amer. Math. Soc., 49 (2012), 475.  doi: 10.1090/S0273-0979-2012-01383-6.  Google Scholar

[21]

L. D. Meeker, Local time-optimal feedback control of strictly normal two-input linear systems,, SIAM journal on control and optimization, 27 (1989), 53.  doi: 10.1137/0327005.  Google Scholar

[22]

Z. Rao and H. Zidani, Hamilton-jacobi-bellman equations on multi-domains,, in Control and Optimization with PDE Constraints, 164 (2013), 93.  doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

[23]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6,, Springer, (1998).  doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[24]

H. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane,, SIAM journal on control and optimization, 25 (1987), 1145.  doi: 10.1137/0325062.  Google Scholar

[25]

M. A. Teixeira, Stability conditions for discontinuous vector fields,, J. Differential Equations, 88 (1990), 15.  doi: 10.1016/0022-0396(90)90106-Y.  Google Scholar

[26]

L. Van den Dries and C. Miller, Geometric categories and o-minimal structures,, Duke Mathematical Journal, 84 (1996), 497.  doi: 10.1215/S0012-7094-96-08416-1.  Google Scholar

show all references

References:
[1]

J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory,, Springer-Verlag New York, (1984).  doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

R. C. Barnard and P. R. Wolenski, Flow invariance on stratified domains,, Set-Valued and Variational Analysis, 21 (2013), 377.  doi: 10.1007/s11228-013-0230-y.  Google Scholar

[3]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, vol. 43,, Springer, (2004).   Google Scholar

[4]

A. Bressan and Y. Hong, Optimal control problems on stratified domains,, Network and Heterogeneous Media, 2 (2007), 313.  doi: 10.3934/nhm.2007.2.313.  Google Scholar

[5]

P. Brunovskỳ, The closed-loop time-optimal control. I: Optimality,, SIAM Journal on Control, 12 (1974), 624.  doi: 10.1137/0312046.  Google Scholar

[6]

P. Brunovskỳ, The closed-loop time optimal control. II: Stability,, SIAM Journal on Control and Optimization, 14 (1976), 156.  doi: 10.1137/0314013.  Google Scholar

[7]

P. Brunovskỳ, Every normal linear system has a regular time-optimal synthesis,, Mathematica Slovaca, 28 (1978), 81.   Google Scholar

[8]

P. Brunovskỳ, Regular synthesis for the linear-quadratic optimal control problem with linear control constraints,, J. Differential Equations, 38 (1980), 344.  doi: 10.1016/0022-0396(80)90012-1.  Google Scholar

[9]

F. Clarke, Discontinuous feedback and nonlinear systems,, in 8th IFAC Symposium on Nonlinear Control Systems, (2010), 1.   Google Scholar

[10]

F. Clarke, Y. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178,, Springer, (1998).   Google Scholar

[11]

A. F. Filippov and F. M. Arscott, Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18,, Springer, (1988).  doi: 10.1007/978-94-015-7793-9.  Google Scholar

[12]

O. Hájek, Terminal manifolds and switching locus,, Mathematical systems theory, 6 (1972), 289.  doi: 10.1007/BF01740720.  Google Scholar

[13]

O. Hájek, Discontinuous differential equations, I,, J. Differential Equations, 32 (1979), 149.  doi: 10.1016/0022-0396(79)90056-1.  Google Scholar

[14]

O. Hájek, Discontinuous differential equations, II,, J. Differential Equations, 32 (1979), 171.   Google Scholar

[15]

C. Hermosilla and H. Zidani, Infinite horizon problem on stratifiable state-constraints set,, J. Differential Equations, 258 (2015), 1430.  doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[16]

S. Honkapohja and T. Ito, Stability with regime switching,, Journal of Economic Theory, 29 (1983), 22.  doi: 10.1016/0022-0531(83)90121-7.  Google Scholar

[17]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields,, SIAM Journal on Applied Dynamical Systems, 8 (2009), 624.  doi: 10.1137/08073113X.  Google Scholar

[18]

V. Y. Kaloshin, A geometric proof of the existence of whitney stratifications,, Mosc. Math. J, 5 (2005), 125.   Google Scholar

[19]

A. Marigo and B. Piccoli, Regular syntheses and solutions to discontinuous odes,, ESAIM: Control, 7 (2002), 291.  doi: 10.1051/cocv:2002013.  Google Scholar

[20]

J. Mather, Notes on topological stability,, Bull. Amer. Math. Soc., 49 (2012), 475.  doi: 10.1090/S0273-0979-2012-01383-6.  Google Scholar

[21]

L. D. Meeker, Local time-optimal feedback control of strictly normal two-input linear systems,, SIAM journal on control and optimization, 27 (1989), 53.  doi: 10.1137/0327005.  Google Scholar

[22]

Z. Rao and H. Zidani, Hamilton-jacobi-bellman equations on multi-domains,, in Control and Optimization with PDE Constraints, 164 (2013), 93.  doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

[23]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6,, Springer, (1998).  doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[24]

H. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane,, SIAM journal on control and optimization, 25 (1987), 1145.  doi: 10.1137/0325062.  Google Scholar

[25]

M. A. Teixeira, Stability conditions for discontinuous vector fields,, J. Differential Equations, 88 (1990), 15.  doi: 10.1016/0022-0396(90)90106-Y.  Google Scholar

[26]

L. Van den Dries and C. Miller, Geometric categories and o-minimal structures,, Duke Mathematical Journal, 84 (1996), 497.  doi: 10.1215/S0012-7094-96-08416-1.  Google Scholar

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[8]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909

[9]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[10]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[11]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[12]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931

[13]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[14]

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

[15]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[16]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

[17]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

[18]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[19]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161

[20]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]