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

Stratified discontinuous differential equations and sufficient conditions for robustness

Cristopher Hermosilla 1,

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.
Keywords: state-constrained ODEs., robustness, existence of solutions, stratified systems, Discontinuous vector fields.
Mathematics Subject Classification: Primary: 34A12, 34A36; Secondary: 34D1.
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]

J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag New York, Inc., 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), 377-403. 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), 313-331. 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), 624-634. 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), 156-162. doi: 10.1137/0314013.  Google Scholar

[7]

P. Brunovskỳ, Every normal linear system has a regular time-optimal synthesis, Mathematica Slovaca, 28 (1978), 81-100.  Google Scholar

[8]

P. Brunovskỳ, Regular synthesis for the linear-quadratic optimal control problem with linear control constraints, J. Differential Equations, 38 (1980), 344-360. 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), 1-29. 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), 289-301. doi: 10.1007/BF01740720.  Google Scholar

[13]

O. Hájek, Discontinuous differential equations, I, J. Differential Equations, 32 (1979), 149-170. doi: 10.1016/0022-0396(79)90056-1.  Google Scholar

[14]

O. Hájek, Discontinuous differential equations, II, J. Differential Equations, 32 (1979), 171-185, 171-185. Google Scholar

[15]

C. Hermosilla and H. Zidani, Infinite horizon problem on stratifiable state-constraints set, J. Differential Equations, 258 (2015), 1430-1460. 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), 22-48. 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), 624-640. doi: 10.1137/08073113X.  Google Scholar

[18]

V. Y. Kaloshin, A geometric proof of the existence of whitney stratifications, Mosc. Math. J, 5 (2005), 125-133.  Google Scholar

[19]

A. Marigo and B. Piccoli, Regular syntheses and solutions to discontinuous odes, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 291-307. doi: 10.1051/cocv:2002013.  Google Scholar

[20]

J. Mather, Notes on topological stability, Bull. Amer. Math. Soc., 49 (2012), 475-506. 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), 53-82. 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, Springer, 164 (2013), 93-116. 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), 1145-1162. doi: 10.1137/0325062.  Google Scholar

[25]

M. A. Teixeira, Stability conditions for discontinuous vector fields, J. Differential Equations, 88 (1990), 15-29. 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), 497-540. 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, Inc., 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), 377-403. 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), 313-331. 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), 624-634. 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), 156-162. doi: 10.1137/0314013.  Google Scholar

[7]

P. Brunovskỳ, Every normal linear system has a regular time-optimal synthesis, Mathematica Slovaca, 28 (1978), 81-100.  Google Scholar

[8]

P. Brunovskỳ, Regular synthesis for the linear-quadratic optimal control problem with linear control constraints, J. Differential Equations, 38 (1980), 344-360. 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), 1-29. 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), 289-301. doi: 10.1007/BF01740720.  Google Scholar

[13]

O. Hájek, Discontinuous differential equations, I, J. Differential Equations, 32 (1979), 149-170. doi: 10.1016/0022-0396(79)90056-1.  Google Scholar

[14]

O. Hájek, Discontinuous differential equations, II, J. Differential Equations, 32 (1979), 171-185, 171-185. Google Scholar

[15]

C. Hermosilla and H. Zidani, Infinite horizon problem on stratifiable state-constraints set, J. Differential Equations, 258 (2015), 1430-1460. 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), 22-48. 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), 624-640. doi: 10.1137/08073113X.  Google Scholar

[18]

V. Y. Kaloshin, A geometric proof of the existence of whitney stratifications, Mosc. Math. J, 5 (2005), 125-133.  Google Scholar

[19]

A. Marigo and B. Piccoli, Regular syntheses and solutions to discontinuous odes, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 291-307. doi: 10.1051/cocv:2002013.  Google Scholar

[20]

J. Mather, Notes on topological stability, Bull. Amer. Math. Soc., 49 (2012), 475-506. 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), 53-82. 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, Springer, 164 (2013), 93-116. 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), 1145-1162. doi: 10.1137/0325062.  Google Scholar

[25]

M. A. Teixeira, Stability conditions for discontinuous vector fields, J. Differential Equations, 88 (1990), 15-29. 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), 497-540. doi: 10.1215/S0012-7094-96-08416-1.  Google Scholar

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