论文标题
关于连续时间线性时间不变系统中可及性的可及性
On the Decidability of Reachability in Continuous Time Linear Time-Invariant Systems
论文作者
论文摘要
我们考虑在连续时间内线性时间不变的控制系统中状态到国家可及性的可决定性。我们分析了有关允许控制集的问题，该问题假定为单位超立方体的线性图下的图像。这种自然模型有界（有时称为饱和）控件。可及性问题的版本的可决定性是$ \ mathbb {r}^n $的仿射子空间是控制理论的基本结果。我们的第一个结果是，如果矩阵$ a $满足某些光谱条件，并且通常是有条件的确定性，则在二维（$ n = 2 $）上的可定义性。如果转换矩阵$ a $是对角线，具有有理条目（或同一代数数的合理倍数），则可及性问题是可决定的。如果转换矩阵$ a $仅具有真实的特征值，则可及性问题是有条件地决定的。时间限制的可及性问题是有条件地决定的，并且在二维中无条件地决定。我们的某些可决定性结果是有条件的，因为它们依赖于某些数学理论的可决定性，即用指数（$ \ Mathfrak {r} _ {\ exp} $）的真实理论以及有界的正弦（$ \ Mathfrak {r} _ {r} _ {\ exp，\ sin} $）。我们还获得了对问题的轻度概括的硬度结果，在该问题中，目标是简单的集合（尺寸$ n-1 $或超平面）而不是一个点，而控制集则是凸有界的多层。在这种情况下，我们表明问题至少与\ emph {连续的阳性问题}或\ emph {非连续连续阳性问题}一样困难。
We consider the decidability of state-to-state reachability in linear time-invariant control systems over continuous time. We analyse this problem with respect to the allowable control sets, which are assumed to be the image under a linear map of the unit hypercube. This naturally models bounded (sometimes called saturated) controls. Decidability of the version of the reachability problem in which control sets are affine subspaces of $\mathbb{R}^n$ is a fundamental result in control theory. Our first result is decidability in two dimensions ($n=2$) if the matrix $A$ satisfies some spectral conditions, and conditional decidablility in general. If the transformation matrix $A$ is diagonal with rational entries (or rational multiples of the same algebraic number) then the reachability problem is decidable. If the transformation matrix $A$ only has real eigenvalues, the reachability problem is conditionally decidable. The time-bounded reachability problem is conditionally decidable, and unconditionally decidable in two dimensions. Some of our decidability results are conditional in that they rely on the decidability of certain mathematical theories, namely the theory of the reals with exponential ($\mathfrak{R}_{\exp}$) and with bounded sine ($\mathfrak{R}_{\exp,\sin}$). We also obtain a hardness result for a mild generalization of the problem where the target is simple set (hypercube of dimension $n-1$ or hyperplane) instead of a point, and the control set is a convex bounded polytope. In this case, we show that the problem is at least as hard as the \emph{Continuous Positivity problem} or the \emph{Nontangential Continuous Positivity problem}.
