The problem of choosing the optimal control of the system, which is described by a parabolic problem with an integral condition over the time and limited internal and starting control, is investigated. The quality criterion will be given by the sum of volume integrals. Using the fundamental solution of the Cauchy problem for the $2b$-parabolic equation, the existence, unity and integral representation of the solutions of the problem for the $2b$-parabolic equation with the integral condition on the time variable were established. Estimates of the solution of the nonlocal problem for the $2b$-parabolic equation with integral condition in time and its derivatives in H$\ddot{\mathrm{o}}$lder spaces are found. The obtained result was used in the study of the problem of optimal control. With the help of the Taylor formula and the integral representation of the solutions of the nonlocal problem, the necessary and sufficient conditions for the existence of the optimal control of the system described by the problem for the $2b$-parabolic equation with the integral condition for the time variable were found.

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