In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is pointwise smaller (in the sense of set containment) than the original set-valued map. The corresponding reduced differential inclusion, defined by the reduced set-valued map, is utilized to develop a generalized notion of a derivative for locally Lipschitz candidate Lyapunov functions in the direction(s) of a set-valued map. The developed generalized derivative yields less conservative statements of Lyapunov stability theorems, invariance theorems, invariance-like results, and Matrosov theorems for differential inclusions. Included illustrative examples demonstrate the utility of the developed theory.
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DOI : 10.1051/cocv/2019074
Keywords: Differential inclusions, stability, hybrid systems, nonlinear systems
@article{COCV_2020__26_1_A24_0,
author = {Kamalapurkar, Rushikesh and Dixon, Warren E. and Teel, Andrew R.},
title = {On reduction of differential inclusions and {Lyapunov} stability},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019074},
mrnumber = {4071314},
zbl = {1441.93202},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019074/}
}
TY - JOUR AU - Kamalapurkar, Rushikesh AU - Dixon, Warren E. AU - Teel, Andrew R. TI - On reduction of differential inclusions and Lyapunov stability JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019074/ DO - 10.1051/cocv/2019074 LA - en ID - COCV_2020__26_1_A24_0 ER -
%0 Journal Article %A Kamalapurkar, Rushikesh %A Dixon, Warren E. %A Teel, Andrew R. %T On reduction of differential inclusions and Lyapunov stability %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019074/ %R 10.1051/cocv/2019074 %G en %F COCV_2020__26_1_A24_0
Kamalapurkar, Rushikesh; Dixon, Warren E.; Teel, Andrew R. On reduction of differential inclusions and Lyapunov stability. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 24. doi: 10.1051/cocv/2019074
[1] and , Differential inclusions. Springer (1984). | MR | Zbl | DOI
[2] and , Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESAIM: COCV 4 (1999) 361–376. | MR | Zbl | Numdam
[3] and , An invariance principle for nonlinear switched systems. Syst. Control Lett. 54 (2005) 1109–1119. | MR | Zbl | DOI
[4] , Discontinuous ordinary differential equations and stabilization. Ph.D. thesis, Universita di Firenze, Italy (1999).
[5] , Optimization and nonsmooth analysis. SIAM (1990). | MR | Zbl | DOI
[6] , Differential equations with discontinuous right-hand sides. Kluwer Academic Publishers (1988). | MR | Zbl | DOI
[7] , and , LaSalle-Yoshizawa corollaries for nonsmooth systems. IEEE Trans. Autom. Control 58 (2013) 2333–2338. | MR | Zbl | DOI
[8] , and , Impulsive and hybrid dynamical systems, Princeton Series in Applied Mathematics (2006). | MR | Zbl
[9] , and , Semistability, finite-time stability, differential inclusions, and discontinuous dynamical systems having a continuum of equilibria. IEEE Trans. Autom. Control 54 (2009) 2465–2470. | MR | Zbl | DOI
[10] , and , On reduction of differential inclusions and Lyapunov stability, in Proc. IEEE Conf. Decis. Control, Melbourne, VIC, Australia (2017) 5499–5504. | Zbl
[11] , and , On reduction of differential inclusions and Lyapunov stability. Preprint (2018). | arXiv | MR
[12] , Nonlinear systems, 3rd edition. Prentice Hall, Upper Saddle River, NJ (2002). | Zbl
[13] and , Game-theoretical control problems. Springer-Verlag, New York (1988). | MR | Zbl
[14] and , Asymptotic behaviour of nonlinear systems. Am. Math. Mon. 111 (2004) 864–889. | MR | Zbl | DOI
[15] , , and , A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems. IEEE Trans. Autom. Control 50 (2005) 183–198. | MR | Zbl | DOI
[16] , On the stability of motion. J. Appl. Math. Mech. 26 (1962) 1337–1353. | Zbl | MR | DOI
[17] and , Qualitative theory of dynamical systems, the role of stability preserving mappings. Marcel Dekker, New York (1995). | MR
[18] and , A chain rule involving vector functions of bounded variation. J. Funct. Anal. 74 (1987) 333–345. | MR | Zbl | DOI
[19] and , Finite time stability of differential inclusions. IMA J. Math. Control Inf . 22 (2005) 465–275. | MR | Zbl | DOI
[20] and , A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans. Circuits Syst. 34 (1987) 73–82. | MR | Zbl | DOI
[21] and , Globally asymptotically stable ‘PD+’ controller for robot manipulators. Int. J. Control 47 (1988) 1697–1712. | Zbl | DOI
[22] and , Vol. 317 of Variational analysis. Springer Science & Business Media (2009). | Zbl
[23] , Stability in general control systems. J. Differ. Equ. 1 (1965) 115–150. | MR | Zbl | DOI
[24] , Principles of mathematical analysis. McGraw-Hill (1976). | MR | Zbl
[25] , Discontinuous feedback and universal adaptive stabilization, in Control of Uncertain systems. Springer (1990) 245–258. | MR | Zbl | DOI
[26] , An integral invariance principle for differential inclusions with applications in adaptive control. SIAM J. Control Optim. 36 (1998) 960–980. | MR | Zbl | DOI
[27] and , Asymptotic stability in hybrid systems via nested Matrosov functions. IEEE Trans. Autom. Control 54 (2009) 1569–1574. | MR | Zbl | DOI
[28] and , Lyapunov stability theory of nonsmooth systems. IEEE Trans. Autom. Control 39 (1994) 1910–1914. | MR | Zbl | DOI
[29] , , and , A refinement of Matrosov’s theorem for differential inclusions. Automatica 68 (2016) 378–383. | MR | Zbl | DOI
[30] , Nonlinear systems analysis, 2nd edition. SIAM (2002). | MR | Zbl | DOI
Cité par Sources :
This research is supported in part by NSF award numbers 1509516 and 1508757, ONR award number N00014-13-1-0151, AFRL award number FA8651-19-2-0009, and AFOSR award number FA9550-15-1-0155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsoring agency.
