On dynamic programming on the values in the semigroup

For not considered previously discrete optimal control problem with target function values in a linearly ordered Abelian semigroup given characterization of the solvability and on its basis the algorithm seeks optimal process with the help of delivering Bellman values elements of limiting sets. We mark the modifications to this algorithm, when

• $P$ is nonempty subset of numbers with the natural ordering and the operation producing the maximum of two numbers;
• $P$ is set of nonnegative numbers with the natural ordering and the addition (or multiplication);
• $P$ is lexicographical product of $m$ (not less than two) linearly ordered Abelian semigroups;
• $P$ is lexicographic product of $m$ (not less than two) sets of real numbers with the natural ordering and the addition, and this algorithm gets $m$-optimal process easier than the previous author's algorithm.