Simultaneous Stabilizability of 3 Linear-systems Is Rationally Undecidable

Blondel, Vincent;Gevers, Michel
(1993) Mathematics of Control, Signals and Systems — Vol. 6, n° 2, p. 135-145 (1993)

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We show that the simultaneous stabilizability of three linear systems, that is the question of knowing whether three linear systems are simultaneously stabilizable, is rationally undecidable. By this we mean that it is not possible to find necessary and sufficient conditions for simultaneous stabilization of the three systems in terms of expressions involving the coefficients of the three systems and combinations of arithmetical operations (additions, subtractions, multiplications, and divisions), logical operations (''and'' and ''or''), and sign test operations (equal to, greater than, greater than or equal to,...).
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Blondel, V., & Gevers, M. (1993). Simultaneous Stabilizability of 3 Linear-systems Is Rationally Undecidable. Mathematics of Control, Signals and Systems, 6(2), 135-145. https://doi.org/10.1007/BF01211744 (Original work published 1993)