Reachability Based Uniform Controllability to Target Set with Evolution Function

In this paper, we investigate the uniform controllability to target set for dynamical systems by designing controllers such that the trajectories evolving from the initial set can enter into the target set. For this purpose, we first introduce the evolution function (EF) for exactly describing the reachable set and give an over-approximation of the reachable set with high precision using the series representation of the evolution function. Subsequently, we propose an approximation approach for Hausdorff semi-distance with a bounded rectangular grid, which can be used to guide the selection of controllers. Based on the above two approximations, we design a heuristic framework to compute a piecewise constant controller, realizing the controllability. Moreover, in order to reduce the computational load, we improve our heuristic framework by the K-arm Bandit Model in reinforcement learning. It is worth noting that both of the heuristic algorithms may suffer from the risk of local optima. To avoid the potential dilemma, we additionally propose a reference trajectory based algorithm for further improvement. Finally, we use some benchmarks with comparisons to show the efficiency of our approach.