On relationship of relative and absolute minimum in synthesis problem

A conventional method for solution of absolute minimum problem in synthesis problem requires knowing of special initial Couchy function for separating from the total integral of the Hamiltonian — Bellman equation a special integral having fixed sign with respect to independent variables of the system. Below there given a method of construction of the above mentioned special integral from $n$-parametric quatient on the basis of which controls are synthesized which ensure the functional absolute minimum. The theory is illustrated with examples.