# American Institute of Mathematical Sciences

• Previous Article
A quantitative internal unique continuation for stochastic parabolic equations
• MCRF Home
• This Issue
• Next Article
A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon
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), 445-471. 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), 1455-1476. doi: 10.1137/S0363012901391676.  Google Scholar [3] F. Ancona and A. Bressan, Nearly time optimal stabilizing patchy feedbacks, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 24 (2007), 279-310. 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, Ser. Adv. Math. Appl. Sci., 76, World Sci. Publ., Hackensack, NJ, 2008, 28-64. 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), 91-99. 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), 87-104.  Google Scholar [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar [8] A. Bressan and F. S. Priuli, Nearly optimal patchy feedbacks, Discr. Cont. Dyn. Systems Series A, 21 (2008), 687-701. 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, R. S. Millman and H. J. Sussmann), Progr. Math., 27, Birkhaüser, Boston, MA, 1983, 181-191.  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, 178, Springer-Verlag, New York, 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), 97-133. 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), 422-441. 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), 747-752. 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), 295-312. 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), 804-833. 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, IEEE Publications, 4, Piscataway, NJ, 1995, 3376-3378. 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), 2103-2117. 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), 205-213. 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), 145-154. 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, Differential Equations, and Control (eds. F. H. Clarke and R. J. Stern), NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Academic, Dordrecht, 1999, 551-598.  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, IEEE Publications, Piscataway, NJ, 1980, 916-921. doi: 10.1109/CDC.1980.271934.  Google Scholar [22] H. J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations, 31 (1979), 31-52. 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), 445-471. 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), 1455-1476. doi: 10.1137/S0363012901391676.  Google Scholar [3] F. Ancona and A. Bressan, Nearly time optimal stabilizing patchy feedbacks, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 24 (2007), 279-310. 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, Ser. Adv. Math. Appl. Sci., 76, World Sci. Publ., Hackensack, NJ, 2008, 28-64. 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), 91-99. 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), 87-104.  Google Scholar [7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar [8] A. Bressan and F. S. Priuli, Nearly optimal patchy feedbacks, Discr. Cont. Dyn. Systems Series A, 21 (2008), 687-701. 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, R. S. Millman and H. J. Sussmann), Progr. Math., 27, Birkhaüser, Boston, MA, 1983, 181-191.  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, 178, Springer-Verlag, New York, 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), 97-133. 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), 422-441. 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), 747-752. 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), 295-312. 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), 804-833. 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, IEEE Publications, 4, Piscataway, NJ, 1995, 3376-3378. 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), 2103-2117. 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), 205-213. 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), 145-154. 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, Differential Equations, and Control (eds. F. H. Clarke and R. J. Stern), NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Academic, Dordrecht, 1999, 551-598.  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, IEEE Publications, Piscataway, NJ, 1980, 916-921. doi: 10.1109/CDC.1980.271934.  Google Scholar [22] H. J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations, 31 (1979), 31-52. doi: 10.1016/0022-0396(79)90151-7.  Google Scholar
 [1] Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1591-1605. doi: 10.3934/dcdss.2020413 [2] Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167 [3] Huawen Ye, Honglei Xu. Global stabilization for ball-and-beam systems via state and partial state feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 17-29. doi: 10.3934/jimo.2016.12.17 [4] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89 [5] Ruofeng Rao, Shouming Zhong. Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1375-1393. doi: 10.3934/dcdss.2020280 [6] Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061 [7] Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control & Related Fields, 2020, 10 (4) : 715-734. doi: 10.3934/mcrf.2020017 [8] Thomas I. Seidman, Houshi Li. A note on stabilization with saturating feedback. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 319-328. doi: 10.3934/dcds.2001.7.319 [9] Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004 [10] Abdallah Benabdallah, Mohsen Dlala. Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $1-d$ heat diffusion equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021092 [11] Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007 [12] Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629-647. doi: 10.3934/mbe.2009.6.629 [13] Tobias Breiten, Karl Kunisch. Boundary feedback stabilization of the monodomain equations. Mathematical Control & Related Fields, 2017, 7 (3) : 369-391. doi: 10.3934/mcrf.2017013 [14] Shui-Hung Hou, Qing-Xu Yan. Nonlinear locally distributed feedback stabilization. Journal of Industrial & Management Optimization, 2008, 4 (1) : 67-79. doi: 10.3934/jimo.2008.4.67 [15] Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 [16] Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks & Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19 [17] Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063 [18] Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085 [19] Imene Aicha Djebour, Takéo Takahashi, Julie Valein. Feedback stabilization of parabolic systems with input delay. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021027 [20] Martin Gugat, Mario Sigalotti. Stars of vibrating strings: Switching boundary feedback stabilization. Networks & Heterogeneous Media, 2010, 5 (2) : 299-314. doi: 10.3934/nhm.2010.5.299

2020 Impact Factor: 1.284