×

Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. (English) Zbl 0927.34034

Stability and stabilizability properties for systems with discontinuous right-hand side by means of locally Lipschitz continuous and regular Lyapunov functions are investigated. The solution of these systems is intended in Filippov’s sense. The stability results are obtained in a more general context of differential inclusions. Concerning stabilizability the authors focus on systems affine with respect to the input. They obtain some sufficient conditions for a system to be stabilized by means of feedback law of Jurdevic-Quinn type.

MSC:

34D20 Stability of solutions to ordinary differential equations
93D15 Stabilization of systems by feedback
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] J.P. Aubin and A. Cellina, Differential Inclusions, Springer Verlag ( 1984). Zbl0538.34007 MR755330 · Zbl 0538.34007
[2] J. Auslander and P. Seibert, Prolongations and Stability in Dynamical Systems. Ann. Inst. Fourier (Grenoble) 14 ( 1964) 237-268. Zbl0128.31303 MR176180 · Zbl 0128.31303 · doi:10.5802/aif.179
[3] A. Bacciotti, Local Stabilizability Theory of Nonlinear System, World Scientific ( 1992). MR1148363 · Zbl 0757.93061
[4] A. Bacciotti and L. Rosier, Liapunov and Lagrange Stability: Inverse Theorems for Discontinuous Systems. Mathematics of Control, Signals and Systems 11 ( 1998) 101-128. Zbl0919.34051 MR1628047 · Zbl 0919.34051 · doi:10.1007/BF02741887
[5] N.P. Bhatia and G.P. Szëgo, Stability Theory of Dynamical Systems, Springer Verlag ( 1970). Zbl0213.10904 MR289890 · Zbl 0213.10904
[6] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley and Sons ( 1983). Zbl0582.49001 MR709590 · Zbl 0582.49001
[7] F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic Controllability Implies Feeedback Stabilization. IEEE Trans. Automat. Control 42 ( 1997) 1394-1407. Zbl0892.93053 MR1472857 · Zbl 0892.93053 · doi:10.1109/9.633828
[8] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski Qualitative Properties of Control Systems: A Survey. J. Dynam. Control Systems 1 ( 1995) 1-47. Zbl0951.49003 MR1319056 · Zbl 0951.49003 · doi:10.1007/BF02254655
[9] 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
[10] K. DeimlingMultivalued Differential Equations, de Gruyter ( 1992). Zbl0760.34002 MR1189795 · Zbl 0760.34002
[11] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC ( 1992). Zbl0804.28001 MR1158660 · Zbl 0804.28001
[12] A.F. Filippov, Differential Equations with Discontinuous Righthandside, Kluwer Academic Publishers ( 1988). Zbl0664.34001 MR1028776 · Zbl 0664.34001
[13] R.A. Freeman and P.V. Kokotovic, Backstepping Design with Nonsmooth Nonlinearities, IFAC NOLCOS, Tahoe City, California ( 1995) 483-488.
[14] V. Jurdjevic and J.P. Quinn, Controllability and Stability. J. Differential Equations 28 ( 1978) 381-389. Zbl0417.93012 MR494275 · Zbl 0417.93012 · doi:10.1016/0022-0396(78)90135-3
[15] O. Hájek, Discontinuous Differential Equations. I, II. J. Differential Equations 32 ( 1979) 149-170, 171-185. Zbl0365.34017 MR534546 · Zbl 0365.34017 · doi:10.1016/0022-0396(79)90056-1
[16] L. Mazzi and V. Tabasso, On Stabilization of Time-Dependent Affine Control Systems. Rend. Sem. Mat. Univ. Politec. Torino 54 ( 1996) 53-66. Zbl0887.93055 MR1490013 · Zbl 0887.93055
[17] E.J. McShane, Integration, Princeton University Press ( 1947). Zbl0033.05302 MR82536 · Zbl 0033.05302
[18] B. Paden and S. Sastry, A Calculus for Computing Filippov’s Differential Inclusion with Application to the Variable Structure Control of Robot Manipulators. IEEE Trans. Circuits and Systems Cas-34 ( 1997) 73-81. Zbl0632.34005 MR871547 · Zbl 0632.34005 · doi:10.1109/TCS.1987.1086038
[19] E.P. Ryan, An Integral Invariance Principle for Differential Inclusions with Applications in Adaptive Control. SIAM J. Control 36 ( 1998) 960-980. Zbl0911.93046 MR1613893 · Zbl 0911.93046 · doi:10.1137/S0363012996301701
[20] D. Shevitz and B. Paden, Lyapunov Stability Theory of Nonsmooth Systems. IEEE Trans. Automat. Control 39 ( 1994) 1910-1914. Zbl0814.93049 MR1293433 · Zbl 0814.93049 · doi:10.1109/9.317122
[21] E.D. Sontag, A Lyapunov-like Characterization of Asymptotic Controllability. SIAM J. Control Optim. 21 ( 1983) 462-471. Zbl0513.93047 MR696908 · Zbl 0513.93047 · doi:10.1137/0321028
[22] E.D. Sontag and H. Sussmann, Nonsmooth Control Lyapunov Functions, Proc. IEEE Conf. Decision and Control, New Orleans, IEEE Publications ( 1995) 2799-2805.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.