Application of Neustadt’s necessary conditions for the optimal control of a linear system with quadratic cost under a closed, hyperplane-bounded half-space state constraint is considered here, Significantly it is proved that the jump-discontinuities generally required in these conditions are not possible except at the initial and final times. This permits simplification of these conditions to a form similar to that obtained for the unconstrained problem of the same type. A non-symmetric matrix Riccati equation is obtained to solve the resulting split-boundary conditions reducing them to a set-valued fixed-point condition on the boundary times. An iterative computational scheme is proposed to solve this fixed-point problem and is applied to two examples.Additionally the existence of the solution to a nonsymmetric Riccati equation of the type obtained here is treated in detail and bounds on this solution are obtained for a large number of general cases.