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Calculus on Manifolds
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
for an arbitrary tensor field T and scalar field f, while the Lie derivative we just introduced instead gives () ℒfXY=[fX,Y]=f[X,Y]−df(Y)X=fℒXY−df(Y)X,
Introduction
Published in M.D.S. Aliyu, Nonlinear H∞-Control, Hamiltonian Systems and Hamilton-Jacobi Equations, 2017
For a smooth function V : ℜn → ℜ, Vx=∂V∂x is the row-vector of first partial-derivatives of V (.) with respect to (wrt) x. Moreover, the Lie-derivative (or directional-derivative) of the function V with respect to a vector-field X is defined as LXV(x)=Vx(x)X(x)=X(V)(x)=∑i=1n∂V∂xi(x1,…,xn)Xi(x1,…,xn).
Supersymmetric Theory of Stochastics:Demystification of Self-Organized Criticality
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
Using the linearity of the Lie derivative in its vector field ℒ^F˜n = ℒ^F + (2T)1/2ξnaℒ^ea,
A hybrid control design for stabilisation of nonlinear systems from the null controllable region*
Published in International Journal of Control, 2023
Maaz Mahmood, Tyler Homer, Prashant Mhaskar
To accommodate non-differentiable Lyapunov functions and discontinuous controllers, we need to introduce the notion of generalised derivatives and gradients. In this work, we utilise Clarke generalised derivatives and gradients (Clarke, 1983). For a locally Lipschitz scalar function , the generalised gradient of V at x is given by where denotes the convex hull of a set and denotes the set of points at which the gradient exists. The Lie derivative of V with respect to a vector field at x is defined by A function is a class- function if it is continuous, strictly increasing and . A scalar function is called proper if it is radially unbounded, i.e. .
Estimates for the volume variation of compact submanifolds driven by a stochastic flow
Published in Dynamical Systems, 2022
Diego Sebastian Ledesma, Robert Andres Galeano Anaya, Fabiano Borges da Silva
Finally, we present below the conditions that we make for most of this work: The vectors fields are smooth (), satisfy and we have the solution flow associated with the SDE (1) for . is a Killing vector field for h, that is, we have that the Lie derivative of h with respect to this vector field is zero: The vectors fields satisfy for all bilinear form .
Disturbance observer-based composite voltage synchronisation control of three-phase four-leg inverter under load variation
Published in International Journal of Control, 2022
Yuge Sun, Chuanlin Zhang, Zhi-gang Su
Some useful notations used throughout this paper are listed as follows. The symbols and denote real number set, positive real number set, the natural number set, the positive integer set, respectively. For integers j and i satisfying , denote as . The symbol denotes the set ratio of all differentiable functions whose first ith time derivatives are continuous. A continuous function is defined by . The Lie derivative is defined by , where is the gradient of , the smooth scalar function is , the vector field is .