For an optimal control problem with a linear differential equation in Hilbert space and quadratic criteria necessary and sufficient conditions of control functions optimality and approximate formulas of the expansions of these functions in eigenfunctions of the control system operator have been obtained.
For the integral operator, which kernel has jump discontinuities on the sides and diagonals of the four equal subsquares of the unit square 0 ≤ x, t ≤ 1, Riesz basisness of its eigen and associated functions is proved.
