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This paper presents necessary and sufficient conditions for the existence of an optimal control for a class of nonlinear systems described by an infinite series of Volterra type. Variational methods are applied to minimize quadratic cost functionals leading to a class of nonlinear integral equations. Sufficient conditions for the existence and uniqueness of a solution of this integral equation are presented using abstract analysis. To the best knowledge of the author these results are new. The technique presented here applies to linear and nonlinear stationary and time-varying systems described by input-output functional relations through finite or infinite Volterra series. An example of nonlinear systems described by an infinite series of Volterra type is presented for illustration.