A theorem of the alternative for SOS Lyapunov functions

2007 ◽  
Author(s):  
Helfried Peyrl ◽  
Pablo A. Parrilo
Keyword(s):  
Lyapunov Functions ◽  
Theorem Of The Alternative

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

Consensus in networks of uncertain robot manipulators without using neighbors’ velocity information

Robotica ◽  
2021 ◽  
pp. 1-17
Author(s):  
Seyed Mostafa Almodarresi ◽  
Marzieh Kamali ◽  
Farid Sheikholeslam
Keyword(s):  
Information Exchange ◽  
Lyapunov Functions ◽  
Robot Manipulators ◽  
Adaptive Methods ◽  
Convergence Time ◽  
Parameter Search ◽  
Velocity Information ◽  
Simulation Results ◽  
Uncertain Robot ◽  
Asymptotic Synchronization

Abstract In this paper, new distributed adaptive methods are proposed for solving both leaderless and leader–follower consensus problems in networks of uncertain robot manipulators, by estimating only the gravitational torque forces. Comparing with the existing adaptive methods, which require the estimation of the whole dynamics, presented methods reduce the excitation levels required for efficient parameter search, the convergence time, and the complexity of the regressor. Additionally, proposed schemes eliminate the need for velocity information exchange between the agents. Global asymptotic synchronization is shown by introducing new Lyapunov functions. Simulation results are provided for a network of 10 4-DOF robot manipulators.

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