Geodesic
On reduction of differential inclusions and Lyapunov stability
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 24.

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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
Classification : 93D02
Keywords: Differential inclusions, stability, hybrid systems, nonlinear systems 
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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. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2019074/

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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.