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Lyapunov's direct method

Module by: Aurèle Balavoine. E-mail the author

Summary: This module describes the direct method of Lyapunov for determining the stability of the equilibrium to an Ordinary Differetial Equation.

Read more about ordinary differential equations, initial value problem and stability of equilibrium.


Determining if an equilibrium of a differential equation is stable or unstable is of great interest in many practical applications. In his memoir [1], Lyapunov developed a method to determine the stability of an equilibrium without having to solve the differential equation nor having to directly apply the definitions of stability. This method, known as Lyapunov's direct method, is related to the notion of energy of the dynamical system.

Let tt be a positive scalar and xx in RNRN a vector of dimension NN. In this section, we are interested in an autonomous ordinary differential equation of the form:

x ˙ ( t ) = f ( x ( t ) ) . x ˙ ( t ) = f ( x ( t ) ) .
(1)

The function f:DRNf:DRN is continuous on DD, a subset of RNRN, and x˙=dx/dtx˙=dx/dt.

Positive Semidefinite function

In order to state the main theorem, we need to introduce the notion of positive definiteness for a function.

Definition 1.1 Let ΩRNΩRN be an open set that contains zero.

A scalar function V:ΩRV:ΩR is positive semidefinite on ΩΩ if it is continuous on ΩΩ and for all xx in ΩΩ, V(x)0V(x)0.

It is positive definite on ΩΩ if for all xx in Ω{0}Ω{0}, V(x)>0V(x)>0.

Similarly, it is negative semidefinite (negative definite) on ΩΩ if -V-V is positive semidefinite (positive definite) on ΩΩ.

Lyapunov's theorem

We can now introduce the theorem of Lyapunov which gives sufficient conditions for an equilibrium to be stable and asymptotically stable.

Theorem 2.1 If there exists an open set ΩΩ that contains zero and a function V:ΩRV:ΩR which is continuous and positive definite on ΩΩ, with V˙(x)0V˙(x)0 for all xΩxΩ (i.e. V˙V˙ is negative semidefinite on ΩΩ), then VV is called a weak Lyapunov function for Equation 1, x(t)=0x(t)=0 is a solution of Equation 1 and this solution is stable.

If, in addition, V˙(x)<0V˙(x)<0 for all xΩ{0}xΩ{0} (i.e. V˙V˙ is negative definite on ΩΩ), then VV is called a Lyapunov function or strict Lyapunov function for Equation 1, and the solution x(t)=0x(t)=0 is asymptotically stable.

Intuitively, the Lyapunov function VV can be interpreted as an energy function for the dynamical system. When the energy in the system decreases with time, the trajectories stop to reach a stable equilibrium that corresponds to a minimum of energy for the system.

Chain rule

The chain rule is an important tool for the study of differential equations. As an example, notice that V˙V˙ can be computed using the chain rule as follows:

V ˙ ( x ( t ) ) = d V ( x ( t ) ) d t = V ( x ) x x ˙ ( t ) = V ( x ) x f ( x ( t ) ) . V ˙ ( x ( t ) ) = d V ( x ( t ) ) d t = V ( x ) x x ˙ ( t ) = V ( x ) x f ( x ( t ) ) .
(2)

As a consequence, the direct method does not require to solve for a solution to the differential equation Equation 1 to determine the stability of the equilibrium.

Extension of the Theorem

Notice that the theorem is stated in the case where x(t)=0x(t)=0 is an equilibrium. In order to determine the stability of any other equilibrium x*x*, it suffices to apply the theorem to the translated differential equation:

u ˙ ( t ) = f ( u ( t ) + x * ) , u ˙ ( t ) = f ( u ( t ) + x * ) ,
(3)

where we set u(t)=x(t)-x*u(t)=x(t)-x*. As a consequence, u(t)=0u(t)=0 is a solution to Equation 3 that has the same stability as x*x*. The theorem can then be applied to the translated Lyapunov function W(t)=V(u(t)+x*)W(t)=V(u(t)+x*) to determine the stability of u(t)=0u(t)=0 or equivalently x(t)=x*x(t)=x*.

Example

Consider the following differential equation, for some αRαR:

x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = - x 1 ( t ) + α x 1 2 ( t ) x 2 ( t ) . x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = - x 1 ( t ) + α x 1 2 ( t ) x 2 ( t ) .
(4)

Consider the following candidate Lyapunov function:

V ( x 1 , x 2 ) = 1 2 ( x 1 2 + x 2 2 ) . V ( x 1 , x 2 ) = 1 2 ( x 1 2 + x 2 2 ) .
(5)

It is clear that VV is continuous and positive definite. We compute its derivative and use the differential equation:

V ˙ ( t ) = x 1 ( t ) x ˙ 1 ( t ) + x 2 ( t ) x ˙ 2 ( t ) = x 1 ( t ) x 2 ( t ) + x 2 ( t ) - x 1 ( t ) + α x 1 2 ( t ) x 2 ( t ) = α x 1 2 ( t ) x 2 2 ( t ) . V ˙ ( t ) = x 1 ( t ) x ˙ 1 ( t ) + x 2 ( t ) x ˙ 2 ( t ) = x 1 ( t ) x 2 ( t ) + x 2 ( t ) - x 1 ( t ) + α x 1 2 ( t ) x 2 ( t ) = α x 1 2 ( t ) x 2 2 ( t ) .
(6)

As a consequence, if α0α0, then VV is negative semidefinite and the equilibrium x*=(0,0)x*=(0,0) is stable. If α<0α<0, then VV is negative definite and the equilibrium x*=(0,0)x*=(0,0) is asymptotically stable. The phase portrait along with a particular solution are plotted on Figure 1 for α=-1α=-1. The particular solution plotted in red indeed converges towards the origin. Also plotted on the figure are the level sets of the Lyapunov function VV, i.e. the contours for which VV is equal to a constant. The value of this constant is given by the color legend on the right side of the figure.

Figure 1: Phase portrait of the differential equation x˙1=x2,x˙2=-x1-x12x2x˙1=x2,x˙2=-x1-x12x2. One solution is plotted in red along with the level sets of the Lyapunov function V=12(x12+x22)V=12(x12+x22).
Figure 1 (lyap.png)

References

  1. Lyapunov, A. M. (1892). The General Problem of the Stability of Motion (In Russian). [translated and edited by A.T. Fuller, London: Taylor & Francis]. Ph. D. Thesis. Univ. Moscow.

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