ⓘ LaSalles invariance principle is a criterion for the asymptotic stability of an autonomous dynamical system. ..

                                     

ⓘ 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