Patchy vector fields and asymptotic stabilization. (English) Zbl 0924.34058

Summary: This paper is concerned with the structure of asymptotically stabilizing feedbacks for a nonlinear control system on. The authors first introduce a family of discontinuous, piecewise smooth vector fields and derive a number of properties enjoyed by solutions to the corresponding ODEs. They define a class of “patchy feedbacks” which are obtained by patching together a locally finite family of smooth controls. The main result shows that, if a system is asymptotically controllable at the origin, then it can be stabilized by a piecewise constant patchy feedback control.


34H05 Control problems involving ordinary differential equations
93D15 Stabilization of systems by feedback
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
Full Text: DOI EuDML


[1] Z. Artstein, Stabilization with relaxed controls. Nonlinear Anal. 7 ( 1983) 1163-1173. Zbl0525.93053 MR721403 · Zbl 0525.93053
[2] A. BacciottiLocal stabilizability of nonlinear control systems. Series on advances in mathematics for applied sciences 8, World Scientific, Singapore ( 1992). Zbl0757.93061 MR1148363 · Zbl 0757.93061
[3] R.W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory, R.W. Brockett, R.S. Millman and H.J. Sussmann, Eds., Birkhauser, Boston ( 1983) 181-191. Zbl0528.93051 MR708502 · Zbl 0528.93051
[4] F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization . IEEE Trans. Automat. Control 42 ( 1997 ) 1394-1407. Zbl0892.93053 MR1472857 · Zbl 0892.93053
[5] F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R.J. Stern, Feedback stabilization and Lyapunov functions , to appear. Zbl0961.93047 MR1780907 · Zbl 0961.93047
[6] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Qualitative properties of trajectories of control systems: A survey . J. Dynamic Control Systems 1 ( 1995) 1-47. Zbl0951.49003 MR1319056 · Zbl 0951.49003
[7] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory 178, Springer-Verlag, New York ( 1998). Zbl1047.49500 MR1488695 · Zbl 1047.49500
[8] G. Colombo, On extremal solutions of differential inclusions. Bull. Polish. Acad. Sci. 40 ( 1992) 97-109. Zbl0771.34017 MR1401862 · Zbl 0771.34017
[9] J.-M. Coron, A necessary condition for feedback stabilization . Systems Control Lett. 14 ( 1990) 227-232. Zbl0699.93075 MR1049357 · Zbl 0699.93075
[10] J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems, Estimation, and Control 4 ( 1994) 67-84. Zbl0925.93827 MR1298548 · Zbl 0925.93827
[11] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift . Math. of Control, Signals, and Systems 5 ( 1992) 295-312. Zbl0760.93067 MR1164379 · Zbl 0760.93067
[12] J.-M. Coron, Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws . SIAM J. Control Optim. 33 ( 1995) 804-833. Zbl0828.93054 MR1327239 · Zbl 0828.93054
[13] J.-M. Coron, L. Praly and A. Teel, Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input-output techniques, in Trends in Control: A European Perspective, A. Isidori, Eds., Springer, London ( 1995) 293-348. MR1448452
[14] A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Acad. Publ. ( 1988). Zbl0664.34001 · Zbl 0664.34001
[15] O. Hájek, Discontinuos differential equations, I-II. J. Differential Equations 32 ( 1979) 149-185. Zbl0365.34017 MR534546 · Zbl 0365.34017
[16] H. Hermes, Discontinuous vector fields and feedback control, in Differential Equations and Dynamical Systems, J.K. Hale and J.P. La Salle, Eds., Academic Press, New York, ( 1967) 155-165. Zbl0183.15905 MR222424 · Zbl 0183.15905
[17] H. Hermes, On the synthesis of stabilizing feedback controls via Lie algebraic methods. SIAM J. Control Optim. 10 ( 1980) 352-361. Zbl0477.93046 MR579546 · Zbl 0477.93046
[18] N.N. Krasovskii and A.I. Subbotin, Positional differential games, Nauka, Moscow, ( 1974) [in Russian]. Revised English translation: Game-theoretical control problems, Springer-Verlag, New York ( 1988). Zbl0298.90067 MR437107 · Zbl 0298.90067
[19] Yu.S. Ledyaev and E.D. Sontag, A remark on robust stabilization of general asymptotically controllable systems, in Proc. Conf. on Information Sciences and Systems (CISS 97), Johns Hopkins, Baltimore, MD ( 1997) 246-251.
[20] Yu.S. Ledyaev and E.D. Sontag, A Lyapunov characterization of robust stabilization. J. Nonlinear Anal, to appear. Zbl0947.34054 MR1695080 · Zbl 0947.34054
[21] S. Nikitin, Piecewise-constant stabilization. SIAM J. Control Optim. to appear. Zbl0922.93043 MR1680818 · Zbl 0922.93043
[22] E.P. Ryan, On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim. 32 ( 1994) 1597-1604. Zbl0806.93049 MR1297100 · Zbl 0806.93049
[23] E.D. Sontag and H.J. Sussmann, Remarks on continuous feedback, in Proc. IEEE Conf. Decision and Control, Aulbuquerque, IEEE Publications, Piscataway ( 1980) 916-921.
[24] E.D. SontagNonlinear regulation: The piecewise linear approach . IEEE Trans. Automat. Control 26 ( 1981) 346-358. Zbl0474.93039 MR613541 · Zbl 0474.93039
[25] E.D. Sontag, Feedback stabilization of nonlinear systems, in Robust Control of Linear Systems and Nonlinear Control, M.A. Kaashoek, J.H. van Shuppen and A.C.M. Ran, Eds., Birkhäuser, Cambridge, MA ( 1990) 61-81. Zbl0735.93063 MR1115377 · Zbl 0735.93063
[26] E.D. Sontag, Mathematical control theory, deterministic finite dimensional systems, Springer-Verlag, New York ( 1990). Zbl0703.93001 MR1070569 · Zbl 0703.93001
[27] E.D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, in Proc. NATO Advanced Study Institute - Nonlinear Analysis, Differential Equations, and Control (Montreal, Jul/Aug 1998), F.H. Clarke and R.J. Stern, Eds., Kluwer ( 1999) 551-598. Zbl0937.93034 MR1695014 · Zbl 0937.93034
[28] H.J. Sussmann, Subanalytic sets and feedback control . J. Differential Equations 31 ( 1979) 31-52. Zbl0407.93010 MR524816 · Zbl 0407.93010
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