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On the Decidability of Reachability in Continuous Time Linear Time-Invariant Systems

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Document pages: 17 pages

Abstract: We consider the decidability of state-to-state reachability in lineartime-invariant control systems over continuous time. We analyse this problemwith respect to the allowable control sets, which are assumed to be the imageunder a linear map of the unit hypercube. This naturally models bounded(sometimes called saturated) controls. Decidability of the version of thereachability problem in which control sets are affine subspaces of$ mathbb{R}^n$ is a fundamental result in control theory. Our first result isdecidability in two dimensions ($n=2$) if the matrix $A$ satisfies somespectral conditions, and conditional decidablility in general. If thetransformation matrix $A$ is diagonal with rational entries (or rationalmultiples of the same algebraic number) then the reachability problem isdecidable. If the transformation matrix $A$ only has real eigenvalues, thereachability problem is conditionally decidable. The time-bounded reachabilityproblem is conditionally decidable, and unconditionally decidable in twodimensions. Some of our decidability results are conditional in that they relyon the decidability of certain mathematical theories, namely the theory of thereals with exponential ($ mathfrak{R} { exp}$) and with bounded sine($ mathfrak{R} { exp, sin}$). We also obtain a hardness result for a mildgeneralization of the problem where the target is simple set (hypercube ofdimension $n-1$ or hyperplane) instead of a point, and the control set is aconvex bounded polytope. In this case, we show that the problem is at least ashard as the emph{Continuous Positivity problem} or the emph{NontangentialContinuous Positivity problem}.

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