Let $\forall k\ge 0. V_k(x)$ be the value function related to the recursive optimization problem
$ J(x_0) = \underset{u}{\inf} \sum_{k=0}^{N-1} x_k^T Q x_k + u_k^T R u_k + x_{N}^T P_N x_N \\ s.t. x_{k+1} = A x_k + B u_k $
then it is possible to prove that
$V_k(x) = \inf_u x^T Q x + u^T R u + V_{k+1}(A x + B u) = x^T P_k x$
where
$ P_k = A^T P_{k+1} A + Q - A^T P_{k+1} B (R + B^T P_{k+1} B)^{-1} B^T P_{k+1} A $
This last equation is the Discrete Algebraic Riccati Equation (DARE). It can be proved that an generalized version of the DARE above is an unique solution for any quadratic dynamic programming functional with linear dynamics such as J(x_0). Therefore the existence of $ J(x_0) $ implies the existence of the DARE, however does the opposite hold?
Does the existence of a DARE implies the existence of a quadratic optimization with linear dynamics?