On the regulator problem for linear systems over rings and algebras

• Main Authors: Hermida-Alonso José Ángel, Carriegos Miguel V., Sáez-Schwedt Andrés, Sánchez-Giralda Tomás
• Format: Article
• Language: English
• Published: De Gruyter 2021-04-01
• Series: Open Mathematics
• Subjects: linear systems over commutative rings; regulator problem; duality principle; pole assignment; 13p25; 93b25
• Online Access: https://doi.org/10.1515/math-2021-0002

The regulator problem is solvable for a linear dynamical system Σ\Sigma if and only if Σ\Sigma is both pole assignable and state estimable. In this case, Σ\Sigma is a canonical system (i.e., reachable and observable). When the ring RR is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).