From 2D system to 1D system via restriction to a vertical strip: applications to time-controllability and zero-time-controllability

Motivated by the role of finite-time restriction in one-dimensional (1D) systems, we investigate the restriction of two-dimensional (2D) systems defined over ${\mathbb{N}}^2$ to a vertical strip. We give a constructive proof that such a restriction induces a 1D system. A Gröbner basis-based algorithm is then developed to compute this restricted 1D system from a given 2D system. The framework is then applied to develop an algorithm for testing the time-controllability and zero-time-controllability of scalar autonomous 2D systems; these properties are known to be equivalent to the existence of a deadbeat controller. Finally, we show that, for time-controllable or zero-time-controllable scalar autonomous 2D systems, the induced 1D system generates all the admissible trajectories of the original 2D system, thereby reducing the complexity of trajectory generation.