• 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 and 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.

[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.

[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.

[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.

[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.

[6]

A. Bressan, Singularities of stabilizing feedbacks, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 87-104.

[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.

[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.

[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.

[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.

[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.

[12]

F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: 10.1137/S036301290240453X.

[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.

[14]

J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.

[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.

[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.

[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.

[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.

[19]

R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbb{R}^{N}$, Nonlinear Anal., 3 (1979), 145-154. doi: 10.1016/0362-546X(79)90044-0.

[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.

[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.

[22]

H. J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations, 31 (1979), 31-52. doi: 10.1016/0022-0396(79)90151-7.

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.

[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.

[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.

[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.

[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.

[6]

A. Bressan, Singularities of stabilizing feedbacks, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 87-104.

[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.

[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.

[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.

[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.

[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.

[12]

F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: 10.1137/S036301290240453X.

[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.

[14]

J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.

[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.

[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.

[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.

[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.

[19]

R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbb{R}^{N}$, Nonlinear Anal., 3 (1979), 145-154. doi: 10.1016/0362-546X(79)90044-0.

[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.

[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.

[22]

H. J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations, 31 (1979), 31-52. doi: 10.1016/0022-0396(79)90151-7.

[1]

Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Control Theory, 2022, 11 (2) : 373-397. 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 and Continuous Dynamical Systems - S, 2022, 15 (5) : 1085-1102. doi: 10.3934/dcdss.2021092

[11]

Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete and 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 and 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 and Management Optimization, 2008, 4 (1) : 67-79. doi: 10.3934/jimo.2008.4.67

[15]

Matthew D. Kvalheim, Daniel E. Koditschek. Necessary conditions for feedback stabilization and safety. Journal of Geometric Mechanics, 2022, 14 (4) : 659-693. doi: 10.3934/jgm.2022013

[16]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control and Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[17]

Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks and Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19

[18]

Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063

[19]

Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085

[20]

Martin Gugat, Mario Sigalotti. Stars of vibrating strings: Switching boundary feedback stabilization. Networks and Heterogeneous Media, 2010, 5 (2) : 299-314. doi: 10.3934/nhm.2010.5.299

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]