Stratified discontinuous differential equations and sufficient conditions for robustness

Cristopher Hermosilla

doi:10.3934/dcds.2015.35.4415

Article Contents

2015, Volume 35, Issue 9: 4415-4437. Doi: 10.3934/dcds.2015.35.4415

This issue Previous Article Robustness of performance and stability for multistep and updated multistep MPC schemes Next Article Dynamic programming using radial basis functions

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

Received: March 2014

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 34A12, 34A36; Secondary: 34D10.

Citation:

Related Papers

Cited by

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.
[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.
[3]	U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, vol. 43, Springer, 2004.
[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.
[5]	P. Brunovskỳ, The closed-loop time-optimal control. I: Optimality, SIAM Journal on Control, 12 (1974), 624-634.doi: 10.1137/0312046.
[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.
[7]	P. Brunovskỳ, Every normal linear system has a regular time-optimal synthesis, Mathematica Slovaca, 28 (1978), 81-100.
[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.
[9]	F. Clarke, Discontinuous feedback and nonlinear systems, in 8th IFAC Symposium on Nonlinear Control Systems, (2010), 1-29.
[10]	F. Clarke, Y. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178, Springer, 1998.
[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.
[12]	O. Hájek, Terminal manifolds and switching locus, Mathematical systems theory, 6 (1972), 289-301.doi: 10.1007/BF01740720.
[13]	O. Hájek, Discontinuous differential equations, I, J. Differential Equations, 32 (1979), 149-170.doi: 10.1016/0022-0396(79)90056-1.
[14]	O. Hájek, Discontinuous differential equations, II, J. Differential Equations, 32 (1979), 171-185, 171-185.
[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.
[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.
[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.
[18]	V. Y. Kaloshin, A geometric proof of the existence of whitney stratifications, Mosc. Math. J, 5 (2005), 125-133.
[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.
[20]	J. Mather, Notes on topological stability, Bull. Amer. Math. Soc., 49 (2012), 475-506.doi: 10.1090/S0273-0979-2012-01383-6.
[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.
[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.
[23]	E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6, Springer, 1998.doi: 10.1007/978-1-4612-0577-7.
[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.
[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.
[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.