The linear-quadratic optimal control problem with positive controllers is considered. Specifically, necessary and sufficient conditions are given for the existence of a solution to the optimal control problem ; the form of these conditions is explicit for the case where a trajectory-dependent term in the integrand of the pay-off functional is absent. Extensive comments are made on the special case of the classical (control-unconstrained) linear-quadratic optimal control problem. In our derivation, methods of non-linear analysis inRnare employed throughout.