Construction of reachability and controllability sets in a special linear control problem

The article considers the problem of constructing reachability and controllability sets for a control problem. The motion of an object is described by a linear system of ordinary differential equations, and control is selected from the class of piecewise-constant functions. Straight boundaries are also set on the controls. The article provides definitions of reachability and controllability sets. It is shown that the problems of constructing these sets are equivalent and can be reduced to the problem of linear mapping of a multidimensional cube. The properties of these sets are also given. In addition, the existing approaches to solving the problem are analyzed. Since they are all too computationally complex, the question of creating a more efficient algorithm arises. The work proposes an algorithm for constructing \newpage the required sets as a system of linear inequalities. A proof of the theorem showing the correctness of the algorithm is provided. The complexity of the presented approach is estimated.