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March  2015, 5(1): 141-163. doi: 10.3934/mcrf.2015.5.141

## State constrained patchy feedback stabilization

 1 Istituto per le Applicazioni del Calcolo “M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, I-00185 Roma, Italy

Received  November 2013 Revised  April 2014 Published  January 2015

We construct a patchy feedback for a general control system on $\mathbb{R}^d$ which realizes practical stabilization to a target set $\Sigma$, when the dynamics is constrained to a given set of states $S$. The main result is that $S$--constrained asymptotically controllability to $\Sigma$ implies the existence of a discontinuous practically stabilizing feedback. Such a feedback can be constructed in patchy'' form, a particular class of piecewise constant controls which ensure the existence of local Carathéodory solutions to any Cauchy problem of the control system and which enjoy good robustness properties with respect to both measurement errors and external disturbances.
Citation: Fabio S. Priuli. State constrained patchy feedback stabilization. Mathematical Control & Related Fields, 2015, 5 (1) : 141-163. doi: 10.3934/mcrf.2015.5.141
##### References:
 [1] F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization,, ESAIM Control Optim. Calc. Var., 4 (1999), 445. doi: 10.1051/cocv:1999117. Google Scholar [2] F. Ancona and A. Bressan, Flow stability of patchy vector fields and robust feedback stabilization,, SIAM J. Control Optim., 41 (2002), 1455. doi: 10.1137/S0363012901391676. Google Scholar [3] F. Ancona and A. Bressan, Nearly time optimal stabilizing patchy feedbacks,, Ann. Inst. Henri Poincaré, 24 (2007), 279. doi: 10.1016/j.anihpc.2006.03.010. Google Scholar [4] F. Ancona and A. Bressan, Patchy feedbacks for stabilization and optimal control: General theory and robustness properties,, in Geometric Control and Nonsmooth Analysis, (2008), 28. doi: 10.1142/9789812776075_0002. Google Scholar [5] A. M. Bloch and S. Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes,, Systems Control Lett., 29 (1996), 91. doi: 10.1016/S0167-6911(96)00049-7. Google Scholar [6] A. Bressan, Singularities of stabilizing feedbacks,, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 87. Google Scholar [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, AIMS Series on Applied Mathematics, (2007). Google Scholar [8] A. Bressan and F. S. Priuli, Nearly optimal patchy feedbacks,, Discr. Cont. Dyn. Systems Series A, 21 (2008), 687. doi: 10.3934/dcds.2008.21.687. Google Scholar [9] R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory (eds. R. W. Brockett, (1983), 181. Google Scholar [10] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Graduate Texts in Mathematics, (1998). Google Scholar [11] F. H. Clarke, L. Rifford and R. J. Stern, Feedback in state constrained optimal control,, ESAIM Control Optim. Calc. Var., 7 (2002), 97. doi: 10.1051/cocv:2002005. Google Scholar [12] F. H. Clarke and R. J. Stern, State constrained feedback stabilization,, SIAM J. Control Optim., 42 (2003), 422. doi: 10.1137/S036301290240453X. Google Scholar [13] F. H. Clarke and R. J. Stern, Lyapunov feedback characterizations state constrained controllability stabilization,, Systems and Control Letters, 54 (2005), 747. doi: 10.1016/j.sysconle.2004.11.013. Google Scholar [14] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signals Systems, 5 (1992), 295. doi: 10.1007/BF01211563. Google Scholar [15] J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback law,, SIAM J. Control Optim., 33 (1995), 804. doi: 10.1137/S0363012992240497. Google Scholar [16] S. Drakunov and V. I. Utkin, Sliding-mode observers: Tutorial,, in Proceedings of the 34th IEEE Conference of Decision and Control, (1995), 3376. doi: 10.1109/CDC.1995.479009. Google Scholar [17] R. Goebel, C. Prieur and A. R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems,, IEEE Trans. Autom. Control, 52 (2007), 2103. doi: 10.1109/TAC.2007.908320. Google Scholar [18] R. Goebel and A. R. Teel, Direct design of robustly asymptotically stabilizing hybrid feedback,, ESAIM Control Optim. Calc. Var., 15 (2009), 205. doi: 10.1051/cocv:2008023. Google Scholar [19] R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbbR^n$,, Nonlinear Anal., 3 (1979), 145. doi: 10.1016/0362-546X(79)90044-0. Google Scholar [20] E. D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances,, in Nonlinear Analysis, (1999), 551. Google Scholar [21] E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback,, in Proceedings of the 19th IEEE Conference on Decision and Control, (1980), 916. doi: 10.1109/CDC.1980.271934. Google Scholar [22] H. J. Sussmann, Subanalytic sets and feedback control,, J. Differential Equations, 31 (1979), 31. doi: 10.1016/0022-0396(79)90151-7. Google Scholar

show all references

##### References:
 [1] F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization,, ESAIM Control Optim. Calc. Var., 4 (1999), 445. doi: 10.1051/cocv:1999117. Google Scholar [2] F. Ancona and A. Bressan, Flow stability of patchy vector fields and robust feedback stabilization,, SIAM J. Control Optim., 41 (2002), 1455. doi: 10.1137/S0363012901391676. Google Scholar [3] F. Ancona and A. Bressan, Nearly time optimal stabilizing patchy feedbacks,, Ann. Inst. Henri Poincaré, 24 (2007), 279. doi: 10.1016/j.anihpc.2006.03.010. Google Scholar [4] F. Ancona and A. Bressan, Patchy feedbacks for stabilization and optimal control: General theory and robustness properties,, in Geometric Control and Nonsmooth Analysis, (2008), 28. doi: 10.1142/9789812776075_0002. Google Scholar [5] A. M. Bloch and S. Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes,, Systems Control Lett., 29 (1996), 91. doi: 10.1016/S0167-6911(96)00049-7. Google Scholar [6] A. Bressan, Singularities of stabilizing feedbacks,, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 87. Google Scholar [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, AIMS Series on Applied Mathematics, (2007). Google Scholar [8] A. Bressan and F. S. Priuli, Nearly optimal patchy feedbacks,, Discr. Cont. Dyn. Systems Series A, 21 (2008), 687. doi: 10.3934/dcds.2008.21.687. Google Scholar [9] R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory (eds. R. W. Brockett, (1983), 181. Google Scholar [10] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Graduate Texts in Mathematics, (1998). Google Scholar [11] F. H. Clarke, L. Rifford and R. J. Stern, Feedback in state constrained optimal control,, ESAIM Control Optim. Calc. Var., 7 (2002), 97. doi: 10.1051/cocv:2002005. Google Scholar [12] F. H. Clarke and R. J. Stern, State constrained feedback stabilization,, SIAM J. Control Optim., 42 (2003), 422. doi: 10.1137/S036301290240453X. Google Scholar [13] F. H. Clarke and R. J. Stern, Lyapunov feedback characterizations state constrained controllability stabilization,, Systems and Control Letters, 54 (2005), 747. doi: 10.1016/j.sysconle.2004.11.013. Google Scholar [14] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signals Systems, 5 (1992), 295. doi: 10.1007/BF01211563. Google Scholar [15] J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback law,, SIAM J. Control Optim., 33 (1995), 804. doi: 10.1137/S0363012992240497. Google Scholar [16] S. Drakunov and V. I. Utkin, Sliding-mode observers: Tutorial,, in Proceedings of the 34th IEEE Conference of Decision and Control, (1995), 3376. doi: 10.1109/CDC.1995.479009. Google Scholar [17] R. Goebel, C. Prieur and A. R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems,, IEEE Trans. Autom. Control, 52 (2007), 2103. doi: 10.1109/TAC.2007.908320. Google Scholar [18] R. Goebel and A. R. Teel, Direct design of robustly asymptotically stabilizing hybrid feedback,, ESAIM Control Optim. Calc. Var., 15 (2009), 205. doi: 10.1051/cocv:2008023. Google Scholar [19] R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbbR^n$,, Nonlinear Anal., 3 (1979), 145. doi: 10.1016/0362-546X(79)90044-0. Google Scholar [20] E. D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances,, in Nonlinear Analysis, (1999), 551. Google Scholar [21] E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback,, in Proceedings of the 19th IEEE Conference on Decision and Control, (1980), 916. doi: 10.1109/CDC.1980.271934. Google Scholar [22] H. J. Sussmann, Subanalytic sets and feedback control,, J. Differential Equations, 31 (1979), 31. doi: 10.1016/0022-0396(79)90151-7. Google Scholar
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