We study linear-quadratic optimal control problems for Voterra systems, and problems that are linear-quadratic in the control but generally nonlinear in the state. In the case of linear-quadratic Volterra control, we obtain sharp necessary and sufficient conditions for optimality. For problems that are linear-quadratic in the control only, we obtain a novel form of necessary conditions in the form of double Volterra equation; we prove the solvability of such equations.
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