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A solution to the problem of ($$A$$,$$B$$)-invariance for series. (English) Zbl 1025.68050
Summary: The aim of this article is to study a standard problem of control theory, the ($$A,B$$)-invariance problem, which amounts to computing a maximal element $$X$$ subject to conditions of the form $$AX\leqslant X+B$$ and $$X\leqslant K$$. We give a solution to the problem in the framework of formal series over particular complete idempotent semirings. Over finite idempotent semirings, we show that, under the assumption that $$B$$ and $$K$$ are recognizable series, the maximal solution exists and is also recognizable. We obtain a similar result for the infinite tropical semiring, with additional hypothesis that the series $$A$$ is a language, but the notion of recognizable series has to be extended to the weaker notion of pseudo-recognizable series.

##### MSC:
 68Q45 Formal languages and automata
##### Keywords:
infinite tropical semiring; pseudo-recognizable series
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##### References:
 [1] Berstel, J.; Reutenauer, C., Rational series and their languages, (1988), Springer Berlin · Zbl 0668.68005 [2] Gunawardena, J., An introduction to idempotency, (), (Chapter 1). · Zbl 0898.16032 [3] Higman, G., Ordering by divisibility in abstract algebras, Proc. London math. soc., 2, 326-336, (1952) · Zbl 0047.03402 [4] Klimann, I., New types of automata to solve fixed point problems, Theoret. comput. sci., 259, 183-197, (2001) · Zbl 0973.68122 [5] Pin, J.-E., Syntactic semigroups, (), (Chapter 10). [6] Pin, J.-E.; Sakarovitch, J., Une application de la représentation matricielle des transductions, Theoret. comput. sci., 35, 271-293, (1985) · Zbl 0563.68064 [7] Sakarovitch, J.; Simon, I., Subwords, (), (Chapter 6). [8] Wonham, W.M., Linear multivariable control—A geometric approach, (1985), Springer Berlin · Zbl 0393.93024
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