I am studying control systems, and I want to solve following problem.

Given full rank state matrix A (with all unstable eigenvalues), design input matrix B, such that cost function J=trace(B′XB) is minimized, where X is the solution to discrete-time Ricatti equation (DARE). I have contraint that (A,B) is stabilizable, i.e.

For a given full rank $A\in {\mathbb{R}}^{n\times n}$, with ${\lambda}_{i}(A)>1$, solve the following

$$\begin{array}{ll}\underset{X\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times m}}{\text{minimize}}& \mathrm{t}\mathrm{r}\left({B}^{\prime}XB\right)\\ \text{subject to}& X={A}^{\prime}X(I+B{B}^{\prime}X{)}^{-1}A\\ & (A,B)\text{is stabilizable}\end{array}$$

From my understanding, since all eigenvalues of A are outside of unit circle (discrete-time system), we can change condition (A,B) is stabilizable with (A,B) is controllable, which is equivalent to rank([$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}([B\phantom{\rule{1em}{0ex}}AB\phantom{\rule{1em}{0ex}}{A}^{2}B\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{A}^{n-1}B])=n$

The problem is for sure feasible, since for any full rank A, there is B such that rank condition is satisfied and we can solve DARE.

Given full rank state matrix A (with all unstable eigenvalues), design input matrix B, such that cost function J=trace(B′XB) is minimized, where X is the solution to discrete-time Ricatti equation (DARE). I have contraint that (A,B) is stabilizable, i.e.

For a given full rank $A\in {\mathbb{R}}^{n\times n}$, with ${\lambda}_{i}(A)>1$, solve the following

$$\begin{array}{ll}\underset{X\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times m}}{\text{minimize}}& \mathrm{t}\mathrm{r}\left({B}^{\prime}XB\right)\\ \text{subject to}& X={A}^{\prime}X(I+B{B}^{\prime}X{)}^{-1}A\\ & (A,B)\text{is stabilizable}\end{array}$$

From my understanding, since all eigenvalues of A are outside of unit circle (discrete-time system), we can change condition (A,B) is stabilizable with (A,B) is controllable, which is equivalent to rank([$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}([B\phantom{\rule{1em}{0ex}}AB\phantom{\rule{1em}{0ex}}{A}^{2}B\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{A}^{n-1}B])=n$

The problem is for sure feasible, since for any full rank A, there is B such that rank condition is satisfied and we can solve DARE.