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Robustness of performance and stability for multistep and updated multistep MPC schemes
Stratified discontinuous differential equations and sufficient conditions for robustness
1. | Project Commands INRIA Saclay & ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762 Palaiseau, France |
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. |
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. |
[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. |
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