TY - GEN
T1 - Higher Order Derivatives of Lyapunov Functions for Stability of Systems with Inputs
AU - Liu, Shenyu
AU - Liberzon, Daniel
N1 - Funding Information:
S. Liu and D. Liberzon are with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, U.S.A. This work was supported by the NSF grant CMMI-1662708 and the AFOSR grant FA9550-17-1-0236.
Publisher Copyright:
© 2019 IEEE.
PY - 2019/12
Y1 - 2019/12
N2 - In this paper we study an alternative method for determining stability of dynamical systems by inspecting higher order derivatives of a Lyapunov function. The system can be time invariant or time varying; in both cases we define the higher order derivatives when there are inputs. We then claim and prove that if there exists a linear combination of those higher order derivatives with non-negative coefficients (except that the coefficient of the 0-th order term needs to be positive) which is negative semi-definite, then the system is globally uniformly asymptotically stable. The proof involves repeated applications of comparison principle for first order differential relations. We also show that a system with inputs whose auxiliary system admits a Lyapunov function satisfying the aforementioned conditions is input-to-state stable.
AB - In this paper we study an alternative method for determining stability of dynamical systems by inspecting higher order derivatives of a Lyapunov function. The system can be time invariant or time varying; in both cases we define the higher order derivatives when there are inputs. We then claim and prove that if there exists a linear combination of those higher order derivatives with non-negative coefficients (except that the coefficient of the 0-th order term needs to be positive) which is negative semi-definite, then the system is globally uniformly asymptotically stable. The proof involves repeated applications of comparison principle for first order differential relations. We also show that a system with inputs whose auxiliary system admits a Lyapunov function satisfying the aforementioned conditions is input-to-state stable.
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U2 - 10.1109/CDC40024.2019.9029302
DO - 10.1109/CDC40024.2019.9029302
M3 - Conference contribution
AN - SCOPUS:85082485146
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 6146
EP - 6151
BT - 2019 IEEE 58th Conference on Decision and Control, CDC 2019
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 58th IEEE Conference on Decision and Control, CDC 2019
Y2 - 11 December 2019 through 13 December 2019
ER -