Гамильтонова связность графов диагональной решетки

A graph $G$ is called Hamiltonian connected graph if for every pair of distinct vertices $u, v \in V(G)$ there exists a hamiltonian $(u,v)$-path in $G$. In this paper we prove Hamiltonian connectivity of the family of infinite two-dimensional diagonal grid induced subgraphs with added horizontal and vertical border edges. A generalization for multidimensional case is given. These results are applied to prove the existence of discrete dynamic systems with arbitrary control functions with some given functioning properties.