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On Recurrent Reachability for Continuous Linear Dynamical Systems

机译：连续线性动力系统的递归可达性

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The continuous evolution of a wide variety of systems, including continuous-time Markov chains and linear hybrid automata, can be described in terms of linear differential equations. In this paper we study the decision problem of whether the solution x(t) of a system of linear differential equations dx/dt = Ax reaches a target halfspace infinitely often. This recurrent reachability problem can equivalently be formulated as the following Infinite Zeros Problem: does a real-valued function f : ℝ≥0 → ℝ satisfying a given linear differential equation have infinitely many zeros? Our main decidability result is that if the differential equation has order at most 7, then the Infinite Zeros Problem is decidable. On the other hand, we show that a decision procedure for the Infinite Zeros Problem at order 9 (and above) would entail a major breakthrough in Diophantine Approximation, specifically an algorithm for computing the Lagrange constants of arbitrary real algebraic numbers to arbitrary precision.
机译：可以用线性微分方程来描述包括连续时间马尔可夫链和线性混合自动机在内的各种系统的连续演化。在本文中，我们研究了线性微分方程组dx / dt = Ax的解x（t）是否经常无限地到达目标半空间的决策问题。该递归可及性问题可以等效地表示为以下无限零问题：实值函数f：ℝ ≥0 →ing满足给定的线性微分方程，它具有无限多个零？我们的主要可判定性结果是，如果微分方程的阶数最多为7，则无限零问题是可判定的。另一方面，我们证明了9级及以上的无穷零问题的决策程序将在Diophantine逼近中取得重大突破，特别是一种用于将任意实数代数的Lagrange常数计算为任意精度的算法。

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