Bellmas's equation of optimization with the periodic control

In this paper the dynamic programming approach is used, and the necessary optimality conditions under a periodic control constraint in a finite interval have been found. On the basis of dynamic programming theory, the optimality conditions were formulated in the form of the modified Bellman's functional equation, which was derived using the method given by Piekarski (1992). The independent variable interval [0,L], is divided into N identical subintervals of the length T=L/N. Let us assume that the control is the same shape within each subintervals. The initial optimization problem within [0,L] is replaced by an equivalent optimization problem valid within one subinterval. Consequently, it is enough to analyze the control function inside a single subinterval, i.e., 0