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We consider the optimal control problem for systems with infinite memory whose models are described by functional-differential equations. We prove the theorem on existence, uniqueness, and continuity of solutions for a system of functional-differential equations in which the delay interval is infinitely large. Sufficient conditions for the existence of optimal controls in the optimal control problems for systems with infinite memory are obtained in terms of the right-hand sides of the equations of motion and the function of performance criterion. [ABSTRACT FROM AUTHOR]

For systems of ordinary differential equations, we obtain some sufficient conditions of existence of an optimal control in terms of their right-hand sides and the cost functional with the use of compactness methods. In the theorems proven, either the time interval is finite and a solution belongs to some domain or the time interval coincides with the semiaxis. [ABSTRACT FROM AUTHOR]