Finding a Lyapunov function for modified quadratic form

Question

I am trying to construct a Lyapunov function to show global asymptotic stability for a somewhat difficult system of equations. I was hoping people might have some suggests on what types of equations to try. The system of $n$ equations is given by:

$\dot{x} = MD(x)Rx-g(x)$

where $M$ is a symmetric matrix, $D(\vec{x})$ is a diagonal matrix with $\vec{x}$ on the diagonals, and $g(x)$ is a positive-definite function that is linear in $x$. If $M$ is the identity then this reduces to a standard quadratic form. But for $M$ otherwise, I am unable to control this system. You can assume that $M$ and $R$ are full rank, and that $M$ is positive definite.

I have been considering equations of the form:

$V(x) = \sum_i (Bx^*)_i \log (\frac{(Bx^*)_i}{(Bx)_i})$

such that:

$\dot{V}(x) = \sum_i \frac{(Bx^*)_i}{(Bx)_i} (B\dot{x})_i$

for some matrix $B$. For example, if $B$ is the identity matrix, then this gives:

$\dot{V}(x) = \sum_i \frac{x^*_i}{x_i} \dot{x}_i$

But I am starting to think that this general form is not an appropriate form to work with, as I cannot make any progress. My idea is that $B$ should be related to $M$ and/or $R$, but no luck.

Are there any obvious things I am missing or alternate Lyapunov functions that I should explore?

Thanks!

Answer 1

I assume you have linearized the system at a particular equilibrium point, which is the origin of your nonlinear differential equation. Note, that we only investigate the stability of equilibrium point of a nonlinear system and not the system as a whole.

If the linearized system is given by

$$\Delta \dot{\boldsymbol{x}}=\boldsymbol{A}\Delta\boldsymbol{x}.$$

Determine all the eigenvalues of $\boldsymbol{A}$. By Lyapunov's indirect method we can distinguish three cases:

1. Case: All eigenvalues have a strictly negative real part. This implies that the equilibrium point of the nonlinear system is at least asymptotically stable.
2. Case: There exists at least one eigenvalue that has a strictly positive real part. This implies instability of the equilibrium point of the nonlinear system.
3. Case: All eigenvalues have a real part that is negative or equal to zero and there are eigenvalues with real part $0$. This is the indecisive case by the indirect method.

If you are in case one then you can invoke Lyapunov's converse theorems. This means that you can use the Lyapunov equation

$$\boldsymbol{PA}+\boldsymbol{A}^T\boldsymbol{P}=-\boldsymbol{Q}$$

in which $\boldsymbol{P}$ is a positive definite symmetric matrix and $\boldsymbol{Q}$ is a positive definite matrix. Often $\boldsymbol{Q}$ is chosen as the identity matrix $\boldsymbol{I}$. By Lyapunov's converse theorem it is guaranteed that there exists a unique $\boldsymbol{P}$ such that

$$V(\Delta\boldsymbol{x})=\Delta\boldsymbol{x}^T\boldsymbol{P}\Delta\boldsymbol{x}$$

is a Lyapunov function of the nonlinear system in a neighbourhood of the equilibrium point (which is shifted to the origin).