]> 8.5 Formal derivation of the optimal state feedback

### 8.5 Formal derivation of the optimal state feedback

The next step in the spectral–factorization method for the lq controller synthesis is ﬁnding the realization of $\stackrel{̂}{\vartheta }\left(\lambda \right)-1$ (where $\stackrel{̂}{\vartheta }$ is the spectral factor deﬁned by (8.13)) as a transfer function of the system similar to (8.7) but with ${c}^{#}$ replaced by a new functional ${g}^{#}$ such that $D\left({g}^{#}\right)\supset \text{C}\left[-r,0\right]\oplus \text{C}\left[-r,0\right]$. To be more precise, we look for a linear continuous functional ${g}^{#}$ on $\text{C}\left[-r,0\right]\oplus \text{C}\left[-r,0\right]$ such that
 ${g}^{#}\mathcal{𝒜}{\left(\mathcal{𝒜}-\lambda I\right)}^{-1}d=\stackrel{̂}{\vartheta }\left(\lambda \right)-1=\frac{\left(p-1\right)+\left(q-b\right){e}^{-2\lambda r}}{1+b{e}^{-2\lambda r}},\phantom{\rule{2em}{0ex}}\lambda \notin \sigma \left(\mathcal{𝒜}\right)$ (8.15)

Now, such linear continuous functional can be represented by the Stieltjes integral

 ${g}^{#}x={\int }_{-r}^{0}d{\Xi }^{\ast }\left(\theta \right)x\left(\theta \right)$ (8.16)

where $\Xi$ is a vector–valued function of bounded variation on $\left[-r,0\right]$. Using (8.16) and (8.10), we can ﬁnd an equivalent representation of (8.15) in terms of entire functions, viz.

 ${\int }_{-r}^{0}d{\Xi }^{\ast }\left(\theta \right){e}^{\lambda \theta }\left[\begin{array}{c}\hfill {e}^{-\lambda r}\hfill \\ \hfill 1\hfill \end{array}\right]=p-1+\left(q-b\right){e}^{-2\lambda r},\phantom{\rule{2em}{0ex}}\lambda \in \mathbb{ℂ}$ (8.17)

Hence, by the injectivity of the ﬁnite Laplace–Stieltjes transform appearing in the left–hand side of (8.17), we obtain

 ${g}^{#}x=\left(p-1\right){x}_{2}\left(0\right)+\left(q-b\right){x}_{1}\left(-r\right),\phantom{\rule{2em}{0ex}}D\left({g}^{#}\right)\supset \text{C}\left[-r,0\right]\oplus \text{C}\left[-r,0\right]$ (8.18)