# Systems of Delay Differential Equations and the Lambert W Function

The evaluation of the Lambert W function for matrices must be understood in order to move forward in our understanding of solving DDE systems. For the DDE system given as

$\dot{\bf x}\left(t\right) = {\bf A}_d{\bf x}\left(t-\tau\right)$

The eigenvalues (in this case we will use $s$ to denote the eigenvalues instead of the more common $\lambda$) of the system can be found as:

$\mbox{det}\left(s{\bf I} - \dfrac{1}{\tau}W_{k}\left({\bf A}_{d}\tau\right)\right) = 0$

Functions of matrices can generally be evaluated using the following relationship

$F\left({\bf A}\right) = {\bf V}F\left({\bf J}\right){\bf V}^{-1}$

where ${\bf A}$ is a matrix, ${\bf V}$ is a matrix of eigenvectors, and ${\bf J}$ is the Jordan canonical form of the matrix ${\bf A}$. The Jordan canonical form of a matrix contains all the eigenvalues of the matrix and accounts for any repeated eigenvalues. As a quick example, consider a matrix ${\bf A}$ with eigenvalues $\lambda_{1}$ and $\lambda_{2}$ and let’s assume that $\lambda_{1}$ is repeated once and $\lambda_{2}$ is repeated twice. This means that, since there are two repeated eigenvalues, there are two Jordan blocks. Then, ${\bf J}$ can be written as

${\bf J} = \left[\begin{array}{ccccc} \lambda_{1} & 1 & 0 & 0 & 0\\ 0 & \lambda_{1} & 0 & 0 & 0\\ 0 & 0 & \lambda_{2} & 1 & 0\\ 0 & 0 & 0 & \lambda_{2} & 1\\ 0 & 0 & 0 & 0 & \lambda_{2} \end{array}\right] = \left[\begin{array}{cc} {\bf J}_{1} & {\bf 0}\\ {\bf 0} & {\bf J}_{2}\end{array}\right]$

${\bf J}_{1} = \left[\begin{array}{cc} \lambda_{1} & 1\\ 0 & \lambda_{1}\end{array}\right]$

${\bf J}_{2} = \left[\begin{array}{ccc} \lambda_{2} & 1 & 0\\ 0 & \lambda_{2} & 1\\ 0 & 0 & \lambda_{2}\end{array}\right]$

Each Jordan block accounts for a single repeated eigenvalue. Now, the expression $F\left({\bf J}\right)$ is

$F\left({\bf J}\right) = \left[\begin{array}{ccccc} F\left(\lambda_{1}\right) & F'\left(\lambda_{1}\right) & 0 & 0 & 0\\ 0 & F\left(\lambda_{1}\right) & 0 & 0 & 0\\ 0 & 0 & F\left(\lambda_{2}\right) & F'\left(\lambda_{2}\right) & \frac{1}{2}F''\left(\lambda_{2}\right)\\ 0 & 0 & 0 & F\left(\lambda_{2}\right) & F'\left(\lambda_{2}\right)\\ 0 & 0 & 0 & 0 & F\left(\lambda_{2}\right) \end{array}\right]$

Thus, in general, the function of a single Jordan block (for a single repeated eigenvalue $\lambda$) can be written as

$F\left({\bf J}\right) = \left[\begin{array}{ccccc} F\left(\lambda\right) & F'\left(\lambda\right) & \frac{1}{2}F''\left(\lambda\right) & \cdots & \frac{1}{\left(n-1\right)!}F^{\left(n-1\right)}\left(\lambda\right)\\ 0 & F\left(\lambda\right) & F'\left(\lambda\right) & \cdots & \frac{1}{\left(n-2\right)!}F^{\left(n-2\right)}\left(\lambda\right)\\ \vdots & \ddots & \ddots & \ddots & \vdots\\ {} & {} & {} & F\left(\lambda\right) & F'\left(\lambda\right)\\ 0 & \cdots & \cdots & \cdots & F\left(\lambda\right) \end{array}\right]$

where $n$ is the duplicity of the eigenvalue. If there are no repeated eigenvalues, the matrix ${\bf J}$ becomes simple diagonal matrix with the eigenvalues on the diagonal (and zeros everywhere else). Therefore, since the Lambert W function is a function (same as sin or cos), the Lambert W of ${\bf A}$ can be written as

$W\left({\bf A}\right) = {\bf V}W\left({\bf J}\right){\bf V}^{-1}$

Differentiating the defining equation $x = W\left(x\right)e^{W\left(x\right)}$ for $W$ and solving for $W'$ (implicit differentiation), we obtain the following expression for the derivative of $W$

$W'\left(x\right) = \dfrac{1}{\left(1 + W\left(x\right)\right)e^{W\left(x\right)}}$
$W'\left(x\right) = \dfrac{W\left(x\right)}{x\left(1 + W\left(x\right)\right)}, \mbox{ for } x\neq 0$

Further derivatives of $W$ can be taken and a general formulation of the $n$th derivative of $W$ can be found in the paper “On the Lambert W Function” by Corless, et al. There is a Matlab function that will evaluate the Lambert W function for matrices (lambertwm), however you must search for it as it was written by a couple of doctoral students at Michigan.

One quick note before we move on to the current problems encountered with DDE systems. For the equation

${\bf Y}e^{\bf Y} = {\bf X}$,

where ${\bf X}$ and ${\bf Y}$ are matrices, the solution, ${\bf Y} = W_{k}\left({\bf X}\right)$, is valid only when ${\bf X}$ and ${\bf Y}$ commute; i.e. when ${\bf XY} = {\bf YX}$. The proof is quite simple and satisfactory. First, assume that ${\bf Y}e^{\bf Y} = {\bf X}$. It follows that

${\bf XY} = {\bf Y}e^{\bf Y}{\bf Y} = {\bf Y}^2e^{\bf Y} = {\bf YX}$

since ${\bf Y}$ will always commute with its own matrix exponential, $e^{\bf Y}$. Thus, it can be seen that the two matrices satisfy the original exponential equation if and only if they commute, and thus satisfying the Lambert W function logically follows.

The next type of system that we will encounter is a DDE system with both a delay term and a non-delay term.

${\bf x}\left(t\right) = {\bf Ax}\left(t\right) + {\bf A}_d{\bf x}\left(t-\tau\right)$

More on this to come. Stay tuned!

## About Jordan Jameson

I am a PhD mechanical engineering student at the University of Maryland College Park. I am performing research at the Center for Advanced Life Cycle Engineering (CALCE) in the area of Prognostics and Health Management (PHM) and physics of failure (POF). I enjoy science and discovery, especially the applications to engineering. Follow me on Twitter @NJordanJameson.

Posted on June 23, 2012, in Engineering, Linear Algebra, Mathematics. Bookmark the permalink. Leave a comment.