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Finding of effective constructive methods of approximate optimal control in feedback form (synthesis) is one of the major problems of the mathematical theory of distributed system optimal control. In the monograph this problem is solved for controlled parabolic and hyperbolic equations with quadratic functional in two situations, which are important for applications. The first one is the case of the presence of fast oscillating coefficients, appearing for modeling of processes in micro-inhomogeneous media. The second one is the case when the equation contains a nonlinear interaction function....