Suppose an LTI discrete-time system is given by the equations $$ x_{k+1} = Ax_k + Bu_k,\\ y_{k} = Cx_k + Du_k $$ with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\rho(A) < 1$ and $x_0 = 0$. Assume $r_k\in\mathbb{R}^{n}$ is a bounded reference for the output sequence. Define the total tracking error as $E(u) := \sum_{k=0}^{\infty}{||r_k - y_k||^2}$ for an input sequence $u := \{u_k\}_{k=0}^{\infty}$.
Question: What are the necessary and sufficient conditions for the existence of a bounded $u^*$ that minimizes $E$? Is output-controllability a sufficient condition?
