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 & 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.  Google Scholar

[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.  Google Scholar

[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.  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 (F. Ancona et al., eds.), World Sci. Publ., Hackensack, NJ, (2008), 28–64. doi: 10.1142/9789812776075_0002.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

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.  Google Scholar

[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.  Google Scholar

[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.  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 (F. Ancona et al., eds.), World Sci. Publ., Hackensack, NJ, (2008), 28–64. doi: 10.1142/9789812776075_0002.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

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