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Totally expanding multiplicative systems. (English) Zbl 1078.15017
A single-matrix multiplicative system is given by the entries of the sequence $$\{Q^{n}x(0)\}_{n=0,1,\dots}$$, where $$Q$$ is an $$N \times N$$ non-negative transition matrix and $$x(0)$$ is an $$N \times 1$$ semi-positive input vector. A single-matrix multiplicative system is said to be totally expanding if each coordinate of the sequence $$(x(1), x(2), \dots)$$ is unbounded.
Here, multiple-matrix multiplicative systems are studied. They are obtained when the single matrix $$Q$$ is replaced by a set $$\{Q^{\delta}: \delta \in D\}$$ of $$N \times N$$ non-negative matrices, where $$D$$ has a “product form” structure $$D= D_1 \times \dots \times D_N$$, where for each $$i$$, $$D_i$$ is a finite non-empty set of non-negative $$1 \times N$$ vectors. Each element $$\delta$$ of $$D$$ is called a policy. Such a system is said to be totally expanding if, for each policy $$\delta$$, each coordinate of the sequence $$(x^{\delta}(1), x^{\delta}(2), \dots)$$ is unbounded.
It is shown that the multiple-matrix multiplicative system is totally expanding if and only if there are no “degenerate” coordinates, the growth rate $$\rho^{\delta}_{i} > 1$$ for each $$\delta$$ and each $$i$$ and there exists an $$N \times 1$$ vector $$u$$ satisfying certain linear inequalities. These linear inequalities can give also the estimate of the smallest coordinate-dependent growth rate of the system.

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15A18 Eigenvalues, singular values, and eigenvectors 90B50 Management decision making, including multiple objectives
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