  In this paper, we study linearly constrained policy optimization over the
manifold of Schur stabilizing controllers, equipped with a Riemannian metric
that emerges naturally in the context of optimal control problems. We provide
extrinsic analysis of a generic constrained smooth cost function, that
subsequently facilitates subsuming any such constrained problem into this
framework. By studying the second order geometry of this manifold, we provide a
Newton-type algorithm that does not rely on the exponential mapping nor a
retraction, while ensuring local convergence guarantees. The algorithm hinges
instead upon the developed stability certificate and the linear structure of
the constraints. We then apply our methodology to two well-known constrained
optimal control problems. Finally, several numerical examples showcase the
performance of the proposed algorithm.
