Abstract: In this article, the problem for the analytical design of the optimum controller is studied by means of the "Second Method" of Liapunov and Bellman’s method of "Dynamic Programming". A sequential approximation method is proposed, it is very convenient both for theoretical analysis and practical calculation.In section 1, a very general statement of the problem is given and it is shown that Bellman’s equation is the basic equation for this problem. In section 2, some basic results related to the Liapunov’s "Second Method" which are necessary for the following study are given in a clearer and simpler manner.In section 3, some results about the problem of the optimum controller are given in general terms, viz: the theorem of uniqueness, the sufficient condition of the optimum problem in terms of Bellman’s equation and the technique of the sequential approximation together with its basic properties.In section 4, the case of the stationary linear system is discussed, the existence and uniqueness problems are solved. Some examples are given in appendix V to explain the method introduced and to show its rapid convergence. It is proved that it converges exponentially.In section 5, two quasi-stationary linear systems are studied, viz: the linear system with slowly-variable coefficients and quasi-linear system. Some interesting examples illustrating some applications of the theoretical results are given in appendix VI.In the last section, some future developments of the problem are discussed.The methods introduced in this paper are all given for the synthesis problem, it is thus very convenient both for theoretical study and for practical application.