December  2018, 11(6): 1103-1119. doi: 10.3934/dcdss.2018063

Linear openness and feedback stabilization of nonlinear control systems

1. 

Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA

2. 

Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA

3. 

Department of Mathematics and Computer Science, Lake Superior State University, Sault Ste. Marie, MI 49783, USA

4. 

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

5. 

RUDN University, Moscow 117198, Russia

Received  March 2017 Revised  July 2017 Published  June 2018

It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variational theory. It allows us, in particular, to obtain efficient conditions via the system data supporting the sufficiency in Brockett's theorem and ensuring local exponential stabilization by means of continuous stationary feedback laws. Furthermore, we derive new necessary conditions for local exponential and asymptotic stabilization of continuous-time control systems by using both continuous and continuously differentiable stationary feedback laws and establish also some counterparts of the obtained sufficient conditions for local asymptotic stabilization by continuous stationary feedback laws in the case of nonlinear discrete-time control systems.

Citation: 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
References:
[1]

S. AdlyA. Hantoute and M. Théra, Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions, Nonlinear Anal., 75 (2012), 985-1008.  doi: 10.1016/j.na.2010.11.009.

[2]

S. AdlyA. Hantoute and M. Théra, Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain, Math. Program., 157 (2016), 349-374.  doi: 10.1007/s10107-015-0938-6.

[3]

F. Alabau-BoussouiraJ.-M. Coron and G. Olive, Internal controllability of first order quasi-linear hyperbolic systems with a reduced number of controls, SIAM J. Control Optim., 55 (2017), 300-323.  doi: 10.1137/15M1015765.

[4]

F. Ancona and A. Bressan, Patchy feedbacks for stabilization and optimal control: General theory and robustness properties, in Geometric Control and Nonsmooth Analysis (F. Ancona et al., eds.), World Sci. Publ., Hackensack, NJ, (2008), 28–64. doi: 10.1142/9789812776075_0002.

[5]

Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173.  doi: 10.1016/0362-546X(83)90049-4.

[6]

J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, Springer, New York, 2005.

[7]

R. W. Brockett, Asymptotic stability and feedback stabilization, In Differential Geometric Control Theory (R. W. Brockett et al., eds.), Birkhäuser, Boston, MA, 27 (1983), 181–191.

[8]

C. I. Byrnes, On Brockett's necessary condition for stabilizability and the topology of Liapunov functions on $\mathbb{R}^N$, Commun. Inf. Syst., 8 (2008), 333-352.  doi: 10.4310/CIS.2008.v8.n4.a1.

[9]

F. H. ClarkeYu. S. LedyaevE. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control, 42 (1997), 1394-1407.  doi: 10.1109/9.633828.

[10]

G. Colombo and K. T. Nguyen, On the minimum time function around the origin, Math. Control Related Fields, 3 (2013), 51-82.  doi: 10.3934/mcrf.2013.3.51.

[11]

J.-M. Coron, A necessary condition for feedback stabilization, Syst. Control Lett., 14 (1990), 227-232.  doi: 10.1016/0167-6911(90)90017-O.

[12]

J.-M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 2007.

[13]

A. V. DmitrukA. A. Milyutin and N. P. Osmolovskii, Lyusternik's theorem and the theory of extrema, Russian Math. Surveys, 35 (1980), 11-51. 

[14]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings: A View from Variational Analysis, Springer, New York, 2014. doi: 10.1007/978-1-4939-1037-3.

[15]

L. M. Graves, Some mapping theorems, Duke Math. J., 17 (1950), 111-114.  doi: 10.1215/S0012-7094-50-01713-3.

[16]

M. L. J. Hautus, Stabilization, controllability and observability for linear autonomous systems, Indagationes Mathematicae (Proceedings), 32 (1970), 448-455. 

[17]

H. Hermes, Asymptotically stabilizing feedback controls and the nonlinear regulator problem, SIAM J. Control Optim., 29 (1991), 185-196.  doi: 10.1137/0329010.

[18]

M. Kawski, Stabilization of nonlinear systems on the plane, Syst. Control Lett., 12 (1989), 169-175.  doi: 10.1016/0167-6911(89)90010-8.

[19]

M. A. Krasnoselskii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis, Moscow, 1975.

[20]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, New York, 1967.

[21]

W. Lin and C. I. Byrnes, Design of discrete-time nonlinear control systems via smooth feedback, IEEE Trans. Autom. Control, 39 (1994), 2340-2346.  doi: 10.1109/9.333790.

[22]

L. A. Lyusternik, On conditional extrema of functionals, Math. Sbornik, 41 (1934), 390-401. 

[23]

B. S. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc., 340 (1993), 1-35.  doi: 10.1090/S0002-9947-1993-1156300-4.

[24]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, Appl. Anal., 90 (2011), 1075-1109.  doi: 10.1080/00036811003735840.

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin, 2006.

[26]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin, 2006.

[27]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[28]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[29]

W. Elmaghraby and P. Keskinocak, Dynamic pricing in the E. D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, In Nonlinear Analysis, Differential Equations and Control (F. H. Clarke and R. J. Stern, eds.), Kluwer, Dordrecht, The Netherlands, 528 (1999), 551–598.

[30]

H. J. SussmannE. D. Sontag and D. Y. Yang, A general result on the stabilization of linear systems using bounded controls, IEEE Trans. Automat. Control, 39 (1994), 2411-2425.  doi: 10.1109/9.362853.

[31]

J. Zabczyk, Some comments on stabilizability, Appl. Math. Optim., 19 (1989), 1-9.  doi: 10.1007/BF01448189.

show all references

References:
[1]

S. AdlyA. Hantoute and M. Théra, Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions, Nonlinear Anal., 75 (2012), 985-1008.  doi: 10.1016/j.na.2010.11.009.

[2]

S. AdlyA. Hantoute and M. Théra, Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain, Math. Program., 157 (2016), 349-374.  doi: 10.1007/s10107-015-0938-6.

[3]

F. Alabau-BoussouiraJ.-M. Coron and G. Olive, Internal controllability of first order quasi-linear hyperbolic systems with a reduced number of controls, SIAM J. Control Optim., 55 (2017), 300-323.  doi: 10.1137/15M1015765.

[4]

F. Ancona and A. Bressan, Patchy feedbacks for stabilization and optimal control: General theory and robustness properties, in Geometric Control and Nonsmooth Analysis (F. Ancona et al., eds.), World Sci. Publ., Hackensack, NJ, (2008), 28–64. doi: 10.1142/9789812776075_0002.

[5]

Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173.  doi: 10.1016/0362-546X(83)90049-4.

[6]

J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, Springer, New York, 2005.

[7]

R. W. Brockett, Asymptotic stability and feedback stabilization, In Differential Geometric Control Theory (R. W. Brockett et al., eds.), Birkhäuser, Boston, MA, 27 (1983), 181–191.

[8]

C. I. Byrnes, On Brockett's necessary condition for stabilizability and the topology of Liapunov functions on $\mathbb{R}^N$, Commun. Inf. Syst., 8 (2008), 333-352.  doi: 10.4310/CIS.2008.v8.n4.a1.

[9]

F. H. ClarkeYu. S. LedyaevE. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control, 42 (1997), 1394-1407.  doi: 10.1109/9.633828.

[10]

G. Colombo and K. T. Nguyen, On the minimum time function around the origin, Math. Control Related Fields, 3 (2013), 51-82.  doi: 10.3934/mcrf.2013.3.51.

[11]

J.-M. Coron, A necessary condition for feedback stabilization, Syst. Control Lett., 14 (1990), 227-232.  doi: 10.1016/0167-6911(90)90017-O.

[12]

J.-M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 2007.

[13]

A. V. DmitrukA. A. Milyutin and N. P. Osmolovskii, Lyusternik's theorem and the theory of extrema, Russian Math. Surveys, 35 (1980), 11-51. 

[14]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings: A View from Variational Analysis, Springer, New York, 2014. doi: 10.1007/978-1-4939-1037-3.

[15]

L. M. Graves, Some mapping theorems, Duke Math. J., 17 (1950), 111-114.  doi: 10.1215/S0012-7094-50-01713-3.

[16]

M. L. J. Hautus, Stabilization, controllability and observability for linear autonomous systems, Indagationes Mathematicae (Proceedings), 32 (1970), 448-455. 

[17]

H. Hermes, Asymptotically stabilizing feedback controls and the nonlinear regulator problem, SIAM J. Control Optim., 29 (1991), 185-196.  doi: 10.1137/0329010.

[18]

M. Kawski, Stabilization of nonlinear systems on the plane, Syst. Control Lett., 12 (1989), 169-175.  doi: 10.1016/0167-6911(89)90010-8.

[19]

M. A. Krasnoselskii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis, Moscow, 1975.

[20]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, New York, 1967.

[21]

W. Lin and C. I. Byrnes, Design of discrete-time nonlinear control systems via smooth feedback, IEEE Trans. Autom. Control, 39 (1994), 2340-2346.  doi: 10.1109/9.333790.

[22]

L. A. Lyusternik, On conditional extrema of functionals, Math. Sbornik, 41 (1934), 390-401. 

[23]

B. S. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc., 340 (1993), 1-35.  doi: 10.1090/S0002-9947-1993-1156300-4.

[24]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, Appl. Anal., 90 (2011), 1075-1109.  doi: 10.1080/00036811003735840.

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin, 2006.

[26]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin, 2006.

[27]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[28]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[29]

W. Elmaghraby and P. Keskinocak, Dynamic pricing in the E. D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, In Nonlinear Analysis, Differential Equations and Control (F. H. Clarke and R. J. Stern, eds.), Kluwer, Dordrecht, The Netherlands, 528 (1999), 551–598.

[30]

H. J. SussmannE. D. Sontag and D. Y. Yang, A general result on the stabilization of linear systems using bounded controls, IEEE Trans. Automat. Control, 39 (1994), 2411-2425.  doi: 10.1109/9.362853.

[31]

J. Zabczyk, Some comments on stabilizability, Appl. Math. Optim., 19 (1989), 1-9.  doi: 10.1007/BF01448189.

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