Reachable block decomposition and same-field state-feedback pole placement

For each integer $k\ge 0$, put ${\mathcal{R}}_k=\mathrm{span}\{A^{j}B{e}_{\mathrm{\ell }}:0\le j<k,1\le \mathrm{\ell }\le m\}$. Then ${\mathcal{R}}_{k+1}={\mathcal{R}}_k+A{\mathcal{R}}_k+\mathrm{im}B$. If ${\mathcal{R}}_{k+1}={\mathcal{R}}_k$, then $\mathrm{im}B\subseteq {\mathcal{R}}_k$ and $A{\mathcal{R}}_k\subseteq {\mathcal{R}}_k$, so every later space equals ${\mathcal{R}}_k$. Otherwise each inclusion is strict; if no equality occurs before ${\mathcal{R}}_n$, its dimension has increased at least $n$ times from zero, hence ${\mathcal{R}}_n=F^n$. Therefore $\mathcal{R}={\mathcal{R}}_n$ is the smallest $A$-invariant subspace containing $\mathrm{im}B$.

Let $(E,H)$ be reachable on $F^d$. If $[\lambda I-EH]$ had rank less than $d$ over $\mathbb{C}$, row-rank equality and rank-nullity applied to its adjoint would give a nonzero $y\in {\mathbb{C}}^d$ with $y^{\ast }(\lambda I-E)=0$ and $y^{\ast }H=0$. Induction then gives $y^{\ast }{E}^{j}H={\lambda }^j{y}^{\ast }H=0$ for every $j<d$, contradicting reachability; for real data the real reachable columns complexify to span ${\mathbb{C}}^d$. Thus the horizontal rank is $d$ for every complex $\lambda$.

Scan the finite columns of $B,AB,\dots ,A^{n-1}B$, retaining a column exactly when it is outside the span of those retained; then scan the standard coordinate vectors until the retained list is a basis of $F^n$. This gives a basis matrix $P$ whose first $r$ columns span $\mathcal{R}$. Invariance and $\mathrm{im}B\subseteq \mathcal{R}$ give $P^{-1}AP=\left[\begin{array}{cc}{A}_r& D\\ 0& A_u\end{array}\right]$ and $P^{-1}B=\left[\begin{array}{c}{B}_r\\ 0\end{array}\right]$; the first $r$ coordinates of the columns $A^{j}B$ are $A_r^j{B}_r$ and span $F^r$. Applying the stabilization argument of step 1.1 in dimension $r$ shows that the first $r$ powers already span $F^r$, so $(A_r,B_r)$ is reachable, including $r=0$. For real data, complexifying the real span gives the complex span because each complex coefficient splits into real and imaginary parts. No arbitrary basis choice is used.

For any gain $J$, a subspace $W$ contains $\mathrm{im}H$ and is $E$-invariant exactly when it contains $\mathrm{im}H$ and is $(E-HJ)$-invariant: $HJW\subseteq \mathrm{im}H\subseteq W$, and the converse follows from $E=(E-HJ)+HJ$. By step 1.1 the smallest invariant subspace containing $\mathrm{im}H$, hence reachability, is feedback-invariant.

In the block coordinates of step 2.1, $[\lambda I-AB]$ has block form $\left[\begin{array}{ccc}\lambda I-A_r& -D& B_r\\ 0& \lambda I-A_u& 0\end{array}\right]$. A left annihilator $(a,b)$ must satisfy $a[\lambda I-A_r{B}_r]=0$, so step 1.2 forces $a=0$; a nonzero remaining annihilator exists exactly when $\det (\lambda I-A_u^{\mathbb{C}})=0$, which by [F5] is exactly $\lambda \in {\sigma }_{\mathbb{C}}(A_u^{\mathbb{C}})$. Invertible coordinate changes preserve rank, proving the claimed equivalence in the original coordinates.

Prove pole placement by induction on $d$. For $d=0$ the empty gain gives determinant $1$. For $d>0$, reachability implies $\mathrm{rank}H>0$. Fix ${\mu }_1\in F$. By step 1.2, $[E-{\mu }_{1}I-H]$ has full row rank over $\mathbb{C}$; for real data a nonzero full-size minor has real nonzero determinant, so it also has full row rank over $F$. Its kernel has dimension $m$ by [F2], whereas kernel vectors with first component zero have dimension $m-\mathrm{rank}H<m$. Choose a kernel vector $(v,u)$ with $v\mathrm{\ne }0$; it satisfies $(E-{\mu }_{1}I)v=Hu$. Choose $v_i\mathrm{\ne }0$ and define $J_{0}x=u{x}_i\mathrm{/}{v}_i$, so $(E-H{J}_0)v={\mu }_{1}v$.

Let $\pi :F^d\to F^d\mathrm{/}Fv$ be the quotient map, ${\bar{E}}_0\pi =\pi (E-H{J}_0)$, and $\bar{H}=\pi H$. The quotient pair is reachable because the spanning columns of $(E-H{J}_0,H)$ project onto the quotient by step 2.2; the stabilization argument of step 1.1 in dimension $d-1$ shows that its first $d-1$ powers suffice. To obtain quotient coordinates over $F$, begin with $v$ and append standard coordinate vectors whenever they increase its span; the resulting finite basis has $d-1$ remaining vectors whose cosets form a basis of $F^d\mathrm{/}Fv$. Thus the induction applies over the original field, also when $d=1$ and the quotient is zero. By induction a quotient gain $\bar{J}$ places ${\mu }_2,\dots ,{\mu }_d$; lift it as $J_1=\bar{J}\pi$ and set $J=J_0+J_1$. Since $J_{1}v=0$, the closed loop retains eigenvalue ${\mu }_1$ on $Fv$ and induces the assigned quotient characteristic polynomial. The block-triangular characteristic identity [F4] gives $\det (zI-E+HJ)={\prod }_{j=1}^d(z-{\mu }_j)$, including repeated poles and poles already in $\sigma (E)$; no inverse of $E$ or $E-\mu I$ is used.