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 H-2 norm of a stable system described by a class of linear time-invariant delay differential equations (DDAEs) with multiple delays. The H-2 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 H-2 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 H-2 norm whenever it is finite and by stating an open problem. (C) 2019 Elsevier Ltd. All rights reserved.