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
The eigenvalues (in this case we will use to denote the eigenvalues instead of the more common ) of the system can be found as:
Functions of matrices can generally be evaluated using the following relationship
where is a matrix, is a matrix of eigenvectors, and is the Jordan canonical form of the matrix . 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 with eigenvalues and and let’s assume that is repeated once and is repeated twice. This means that, since there are two repeated eigenvalues, there are two Jordan blocks. Then, can be written as
Each Jordan block accounts for a single repeated eigenvalue. Now, the expression is
Thus, in general, the function of a single Jordan block (for a single repeated eigenvalue ) can be written as
where is the duplicity of the eigenvalue. If there are no repeated eigenvalues, the matrix 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 can be written as
Differentiating the defining equation for and solving for (implicit differentiation), we obtain the following expression for the derivative of
Further derivatives of can be taken and a general formulation of the th derivative of 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
where and are matrices, the solution, , is valid only when and commute; i.e. when . The proof is quite simple and satisfactory. First, assume that . It follows that
since will always commute with its own matrix exponential, . 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.
More on this to come. Stay tuned!