Analyzing ultimate positivity for solvable systems

The positivity problem is a foundational decision problem. It asks whether a dynamical system would keep the observing expression (over its states) positive. It has a derivative-the ultimate positivity problem, which allows that the observing expression is non-positive within a bounded time interval. For the two problems, most existing results are established on discrete-time dynamical systems, specifically on linear recurrence sequences. In this paper, however, we study the ultimate positivity problem for a class of continuous-time dynamical systems, called solvable systems. They subsume linear systems. For the general solvable system, we present a sufficient condition for inferring ultimate positivity. The validity of the condition can be algorithmically checked. Once it is valid, we can further find the time threshold, after which the observing expression would be always positive. On the other hand, we show that the ultimate positivity problem is decidable for some special classes of solvable systems, such as linear systems of dimension up to five.