In this paper, we consider the optimal control problem in the feedback form (synthesis) for a parabolic equation with rapidly oscillating coefficients and not-decomposable quadratic cost functional with superposition type operator. In general, it is not possible to find the exact formula of optimal synthesis for such a problem because the Fourier method can't be directly applied. But transition to the homogenized parameters greatly simplifies the structure of the problem. Assuming that the problem with the homogenized coefficients already admits optimal synthesis form, we ground approximate optimal control in the feedback form for the initial problem. We give an example of superposition operator for specific conditions in this paper.

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In this paper, we use the averaging method to find an approximate solution in the optimal control problem of a parabolic system with non-linearity of the form f(t/ε,y) on an infinite time interval.

In this paper, we develop a general approach to investigate limit dynamics of infinite-dimensional dissipative impulsive systems whose initial conditions do not uniquely determine their long time behavior. Based on the notion of an uniform attractor, we show how to describe limit behavior of such complex systems with the help of properties of their components. More precisely, we prove the existence of the uniform attractor for an impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness.

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