On the m-dimensional Cayley–Hamilton theorem and its application to an algebraic decision problem inferred from the H2 norm analysis of delay systems

We consider a recursion formula for multi-dimensional powers of a finite set of matrices, which can be interpreted as a natural generalization of the celebrated Cayley–Hamilton theorem, and we show how it allows to solve an algebraic decision problem on a semigroup of matrices, which bears similarities to the observability problem of a switched linear system. This problem appears in the computation of the norm of a stable system described by a class of linear time-invariant delay differential equations (DDAEs) with multiple delays. The norm of a DDAE may not be finite even if there are seemingly no direct feedthrough terms. We show that necessary and sufficient conditions for a finite norm consist of an infinite number of linear equations to be satisfied, inducing the algebraic decision problem, and that using the generalized Cayley–Hamilton theorem checking these conditions can be turned into a check of a finite number of equations. We conclude with some comments on the computation of the norm whenever it is finite and by stating an open problem