LaSalle's invariance principle

On a compact positively invariant set where

\dot{V} \leq 0

, every trajectory approaches the largest invariant subset of

{\dot{V} = 0}

.

Table of contents
- Damped mechanical energy as a Lyapunov function LaSalle's invariance principle
  - Damped rational double-barrier phase portrait Damped mechanical energy as a Lyapunov function
    - Outward escape from the rational potential barrier Damped rational double-barrier phase portrait

Damped mechanical energy as a Lyapunov function

For

\overset{x}{¨} = - U^{'} (x) - μ \overset{x}{˙}

with

μ > 0

,

E (x, \overset{x}{˙}) = \frac{1}{2} \overset{x}{˙}^{2} + U (x), \dot{E} = - μ \overset{x}{˙}^{2} .

If

U

is radially unbounded, every trajectory is bounded, and LaSalle's principle reduces its omega-limit set to invariant points with zero velocity.

Damped rational double-barrier phase portrait

For

\overset{x}{¨} + k \overset{x}{˙} + \frac{2 x ( 1 - x ^{2} )}{( 1 + x ^{2} ) ^{3}} = 0,

the mechanical energy is

E (x, y) = \frac{1}{2} y^{2} + \frac{x ^{2}}{( 1 + x ^{2} ) ^{2}}, \dot{E} = - k y^{2} .

The origin is a center for

k = 0

and an asymptotically stable equilibrium for

k > 0

, while

(\pm 1, 0)

are saddles. For positive damping, their stable manifolds form the boundary between the basin of the origin and the two escape regions.

Outward escape from the rational potential barrier

For

k > 0

, the trajectory beginning at

(x, y) = (1, y_{0})

with

y_{0} > 0

remains in

x > 1

,

y > 0

and obeys

0 < y (t) < y_{0}^{2} + \frac{1}{2} .

It enters every strip

0 < y < ε

: otherwise

\dot{E} = - k y^{2} \leq - k ε^{2}

would eventually make the nonnegative energy negative.

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Past exam of the mathematics course of the University of Cambridge / 2001 / ia / Paper 2 / 8B / Solution

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LaSalle's invariance principle is a fundamental result in the field of dynamical systems and control theory that provides conditions under which the behavior of a dynamical system can be analyzed in terms of its invariant sets. It is particularly useful in the study of stability for nonlinear systems.

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