Suppose I have a discrete-time linear dynamical system with no assumptions made about its properties e.g. stability, controllability, reachability. I have an arbitrary target $x^*$ chosen from its controllable subspace. Is it guaranteed that I can always design an LQR controller of the form $$J = (x_T - x^*)^TF(x_T - x^*) + \sum_{t=1}^{T-1} (x_t - x^*)^TQ(x_t - x^*) + u_t^T R u_t$$ to exactly reach the target? Perhaps with some restrictions on $Q,R,F,T$? Or, does controllability of $x^*$ only imply that there is some sequence of $u_t$'s, which may or may not come from an LQR formulation?
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