The article investigates the solvability of the problem of synthesis, distributed and boundary control while minimizing the piecewise linear functional, in the case of control of oscillatory processes described by integro-differential partial differential equations with the Fredholm integral operator. The functions of external and boundary influences are nonlinear with respect to controls. An integro-differential equation of a specific type is obtained for the Bellman functional. The algorithm for constructing a solution to the problem of synthesis, distributed and boundary controls is described, the procedure for determining equations as functions (functionals) of the state of the controlled process is described.