ⓘ LaSalles invariance principle
LaSalles invariance principle is a criterion for the asymptotic stability of an autonomous dynamical system.
1. Global version
Suppose a system is represented as
x = f x {\displaystyle {\dot {\mathbf {x} }}=f\left\mathbf {x} \right}where x {\displaystyle \mathbf {x} } is the vector of variables, with
f 0 = 0. {\displaystyle f\left\mathbf {0} \right=\mathbf {0}.}If a C 1 {\displaystyle C^{1}} function V x {\displaystyle V\mathbf {x}} can be found such that
V x ≤ 0 {\displaystyle {\dot {V}}\mathbf {x}\leq 0} for all x {\displaystyle \mathbf {x} } negative semidefinite,then the set of accumulation points of any trajectory is contained in I {\displaystyle {\mathcal {I}}} where I {\displaystyle {\mathcal {I}}} is the union of complete trajectories contained entirely in the set { x: V x = 0 } {\displaystyle \{\mathbf {x}:{\dot {V}}\mathbf {x}=0\}}.
If we additionally have that the function V {\displaystyle V} is positive definite, i.e.
V x > 0 {\displaystyle V\mathbf {x}> 0}, for all x ≠ 0 {\displaystyle \mathbf {x} \neq \mathbf {0} } V 0 = 0 {\displaystyle V\mathbf {0}=0}and if I {\displaystyle {\mathcal {I}}} contains no trajectory of the system except the trivial trajectory x t = 0 {\displaystyle \mathbf {x} t=\mathbf {0} } for t ≥ 0 {\displaystyle t\geq 0}, then the origin is asymptotically stable.
Furthermore, if V {\displaystyle V} is radially unbounded, i.e.
V x → ∞ {\displaystyle V\mathbf {x}\to \infty }, as ‖ x ‖ → ∞ {\displaystyle \Vert \mathbf {x} \Vert \to \infty }then the origin is globally asymptotically stable.
2. Local version
If
V x > 0 {\displaystyle V\mathbf {x}> 0}, when x ≠ 0 {\displaystyle \mathbf {x} \neq \mathbf {0} } V x ≤ 0 {\displaystyle {\dot {V}}\mathbf {x}\leq 0}hold only for x {\displaystyle \mathbf {x} } in some neighborhood D {\displaystyle D} of the origin, and the set
{ V x = 0 } ⋂ D {\displaystyle \\mathbf {x}=0\}\bigcap D}does not contain any trajectories of the system besides the trajectory x t = 0, t ≥ 0 {\displaystyle \mathbf {x} t=\mathbf {0},t\geq 0}, then the local version of the invariance principle states that the origin is locally asymptotically stable.
3. Relation to Lyapunov theory
If V x {\displaystyle {\dot {V}}\mathbf {x}} is negative definite, the global asymptotic stability of the origin is a consequence of Lyapunovs second theorem. The invariance principle gives a criterion for asymptotic stability in the case when V x {\displaystyle {\dot {V}}\mathbf {x}} is only negative semidefinite.
4. Example: the pendulum with friction
This section will apply the invariance principle to establish the local asymptotic stability of a simple system, the pendulum with friction. This system can be modeled with the differential equation Clearly, V x 1, x 2 {\displaystyle Vx_{1},x_{2}} is positive definite in an open ball of radius π {\displaystyle \pi } around the origin. Computing the derivative,
V x 1, x 2 = g l sin x 1 x 1 + x 2 x 2 = − k m x 2 {\displaystyle {\dot {V}}x_{1},x_{2}={\frac {g}{l}}\sin x_{1}{\dot {x}}_{1}+x_{2}{\dot {x}}_{2}={\frac {k}{m}}x_{2}^{2}}Observe that V 0 = V 0 = 0 {\displaystyle V0={\dot {V}}0=0}. If it were true that V < 0 {\displaystyle {\dot {V}}
 contributions to stability theory, such as LaSalle s invariance principle which bears his name. Joseph La Salle defended his Ph.D. thesis on Pseudo  Normed
 Records La Salle Theater Chicago Jones Lang La Salle a financial and professional services company specializing in real estate LaSalle s invariance principle
 modern life as well as Joseph P. La Salle who in 1960 published a paper describing LaSalle s invariance principle Jack K. Hale, Harold J. Kushner and
 de La Salle 1643 1687 French explorer of North America Joseph P. La Salle 1916 1983 American mathematician known for the La Salle invariance principle
 symmetry includes invariance principles that have been used in some discrete approaches to quantum gravity where the diffeomorphism invariance of general relativity
 the error converges. Lyapunov function Perturbation theory LaSalle s invariance principle Lyapunov, A. M. The General Problem of the Stability of Motion
 Landauer s principle there is a minimum possible amount of energy required to change one bit of information, known as the Landauer limit. LaSalle s invariance
 it makes no mention of simplicity. It shares with the special principle the invariance of the form of the description among mutually translating reference
 including gravitational waves by extending the validity of Lorentz  invariance to non  electrical forces. Eventually Poincare independently of Einstein
 omitted. Physics portal Science portal Doubly special relativity Galilean invariance General relativity references Scale relativity Special relativity references
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