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Solvability of start control problems for semilinear distributed Sobolev type systems

In this paper the existence of solutions of start control problem for the system described by first order semilinear equation in Hilbert space with degenerate operator at derivative is researched. For this purpose, sufficient conditions for the existence of a unique solution of the Cauchy problem for the equation are found. The solvability of the start control problem is proved under the assumption of the relative sectoriality of linear operators in the equation, some minimal assumptions for non-linear operator and the condition of the existence of at least one solution for the corresponding Cauchy problem on given interval. Abstract result is illustrated by an example of start control problem for semilinear system of partial integro-differential equations.