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One way to express correctness of a Petri net N is to specify a linear inequality U, requiring each reachable marking of N to satisfy U. A linear inequality U is stable if it is preserved along steps. If U is stable, then verifying correctness reduces to checking U in the initial marking of N. In this paper, we characterize classes of stable linear inequalities of a given Petri net by means of structural properties. We generalize classical results on traps, co-traps, and invariants. We show how to decide stability of a given inequality. For a certain class of inequalities, we present a polynomial time decision procedure. Furthermore, we show that stability is a local property and exploit this for the analysis of asynchronously interacting open net structures. Finally, we study the summation of inequalities as means of deriving valid inequalities.
We consider the following problems. Given a discrete-time linear system, find, if possible, linear state-feedback control laws such that the corresponding closed-loop system trajectory is positive whenever the initial state is positive. This problem is called the feedback holdability problem. If, in addition, the requirement of non-negativity is imposed on controls, the problem is a positive feedback holdability problem. In the paper, necessary and sufficient conditions for feedback and positive feedback holdability are established in a form of systems of linear inequalities and a procedure for computing the set of all state-feedback controllers that make the closed-loop system holdable is proposed. The relation between controllability and holdability is also treated. Feedback and positive feedback holdability of the class of positive systems is considered as well.
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