if there were only the last two terms on the LHS, then maybe a Scuir complement
[c*eye(N) w’;w u] >= 0 (in sdp mode)
But I’m not sure that can be extended to deal with the terms in the sum, because I think that would invalidate the Schur complement. Maybe someone else knows how to do that.
So your first task is to prove that this is a convex constraint, and specifically, that it can be formulated as a LINEAR SDP