]> 3.2 Evaluation of the quadratic integrals

### 3.2 Evaluation of the quadratic integrals

Motivated by examples of Section 3.1, we pose the problem of evaluating the quadratic integral performance index
 $J\left(\left[\begin{array}{c}\hfill {v}_{0}\hfill \\ \hfill \phi \hfill \end{array}\right]\right)={\int }_{0}^{\infty }\left[{v}^{T}\left(t\right),{z}^{T}\left(t-r\right)\right]\left[\begin{array}{cc}P\hfill & Q\hfill \\ {Q}^{T}\hfill & R\hfill \end{array}\right]\left[\begin{array}{c}\hfill v\left(t\right)\hfill \\ \hfill z\left(t-r\right)\hfill \end{array}\right]dt$ (3.16)

with $P$, $Q$, $R\in L\left({\mathbb{ℝ}}^{n}\right)$, $P={P}^{T}$, $R={R}^{T}$, $\left[\begin{array}{cc}P\hfill & Q\hfill \\ {Q}^{T}\hfill & R\hfill \end{array}\right]\ge 0$over trajectories of the neutral system (3.5). We shall give a solution to this problem employing the results of Section 2.4.

In the state space $\text{H}={\mathbb{𝕄}}^{2}={\mathbb{ℝ}}^{n}\oplus {\text{L}}^{2}\left(-r,0;{\mathbb{ℝ}}^{n}\right)$ we can write (3.5) as an abstract initial value problem appearing in (2.19) with

 $\left\{\begin{array}{ccc}\hfill \mathcal{𝒜}x& \hfill =\hfill & \mathcal{𝒜}\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]=\left[\begin{array}{c}\hfill {A}_{1}v+\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)\hfill \\ \hfill {\psi }^{\prime }\hfill \end{array}\right]\hfill \\ \hfill D\left(\mathcal{𝒜}\right)& \hfill =\hfill & \left\{\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]\in {\mathbb{ℝ}}^{n}\oplus {\text{W}}^{1,2}\left(-r,0;{\mathbb{ℝ}}^{n}\right),\phantom{\rule{0ex}{0ex}}v=\psi \left(0\right)-{A}_{0}\psi \left(-r\right)\right\}\hfill \end{array}\right\}$ (3.17)

and with and initial point ${x}_{0}=\left[\begin{array}{c}\hfill {v}_{0}\hfill \\ \hfill \phi \hfill \end{array}\right]$. We shall prove that $\mathcal{𝒜}$ generates a linear ${\text{C}}_{0}$–semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ on $\text{H}$,

$S\left(t\right)\left[\begin{array}{c}\hfill {v}_{0}\hfill \\ \hfill \phi \hfill \end{array}\right]=\left[\begin{array}{c}\hfill v\left(t\right)\hfill \\ \hfill {z}_{t}\hfill \end{array}\right],\phantom{\rule{2em}{0ex}}t\ge 0$

where

${z}_{t}:\phantom{\rule{0ex}{0ex}}\left[-r,0\right]\ni \theta ↦{z}_{t}\left(\theta \right)=z\left(t+\theta \right)\in {\mathbb{ℝ}}^{n}.$

To do this the following result will be useful.

Theorem 3.2.1 (Walker). Let $\text{H}$ be a real Hilbert space with scalar product ${〈\cdot ,\cdot 〉}_{\text{H}}$. Assume that $\mathcal{𝒜}:\phantom{\rule{0ex}{0ex}}\left(D\left(\mathcal{𝒜}\right)\subset \text{H}\right)\to \text{H}$ is a linear operator satisfying the assumptions:

(i)
there exists ${\mu }_{0}>0$ such that $R\left(\mu I-\mathcal{𝒜}\right)=\text{H}$ for all $\mu >{\mu }_{0}$,
(ii)
the exist $\omega \in \mathbb{ℝ}$ and an equivalent scalar product ${〈\cdot ,\cdot 〉}_{e}$ in $\text{H}$ such that
${〈x,\mathcal{𝒜}x〉}_{e}\le \omega {∥x∥}_{e}^{2}\phantom{\rule{2em}{0ex}}\forall x\in D\left(\mathcal{𝒜}\right).$

Then $\mathcal{𝒜}$ generates a ${\text{C}}_{0}$–semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ on $\text{H}$ with the property

${∥S\left(t\right){x}_{0}∥}_{e}\le {e}^{\omega t}{∥{x}_{0}∥}_{e}\phantom{\rule{2em}{0ex}}\forall {x}_{0}\in \text{H}\phantom{\rule{1em}{0ex}}\forall t\ge 0.$

Proof. Recall that a scalar product ${〈\cdot ,\cdot 〉}_{e}$ is equivalent with the original scalar product ${〈\cdot ,\cdot 〉}_{\text{H}}$ if the norms induced by these scalar products are equivalent, i.e., there exist positive constants ${c}_{1}$, ${c}_{2}$ such that

${c}_{1}{∥x∥}_{\text{H}}\le {∥x∥}_{e}\le {c}_{2}{∥x∥}_{\text{H}}\phantom{\rule{2em}{0ex}}\forall x\in \text{H}.$

The proof relies on verifying all assumptions of Theorem 2.3.2 as a sufficient condition for generation of the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$. Details can be found in [84, Theorem 4.2, p. 108]. □

Observe that the operator (3.17) satisﬁes condition (i) of Theorem 3.2.1 if for sufficiently large $\mu >0$ the equation

$\mu \left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]-\mathcal{𝒜}\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]=\left[\begin{array}{c}\hfill \stackrel{̃}{v}\hfill \\ \hfill \stackrel{̃}{\psi }\hfill \end{array}\right]\in \text{H}$

has a solution in $D\left(\mathcal{𝒜}\right)$. Equivalently, we seek for a solution of the system

$\left\{\begin{array}{ccc}\hfill \mu \left[\psi \left(0\right)-{A}_{0}\psi \left(-r\right)\right]-{A}_{1}\psi \left(0\right)-{A}_{2}\psi \left(-r\right)& \hfill =\hfill & \stackrel{̃}{v}\hfill \\ \hfill \mu \psi \left(\theta \right)-{\psi }^{\prime }\left(\theta \right)& \hfill =\hfill & \stackrel{̃}{\psi }\left(\theta \right)\hfill \end{array}\right\}$

satisfying $\psi \in {\text{W}}^{1,2}\left(-r,0;{\mathbb{ℝ}}^{n}\right)$. Solving the second equation and substituting the solution into the ﬁrst equation, we obtain a nonhomogeneous algebraic linear equation in ${\mathbb{ℝ}}^{n}$,

$\mu \left[I-\frac{1}{\mu }{A}_{1}-\frac{1}{\mu }{e}^{-\mu r}{A}_{2}-{e}^{-\mu r}{A}_{0}\right]\psi \left(0\right)={e}^{-\mu r}\left(\mu {A}_{0}+{A}_{2}\right){\int }_{r}^{0}\stackrel{̃}{\psi }\left(\tau \right){e}^{-\mu \tau }d\tau +\stackrel{̃}{v},$

which has a solution because

${∥\frac{1}{\mu }{A}_{1}+\frac{1}{\mu }{e}^{-\mu r}{A}_{2}+{e}^{-\mu r}{A}_{0}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}\le$

$\le \frac{1}{\mu }\left[{∥{A}_{1}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}+{∥{A}_{2}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}+\frac{1}{re}{∥{A}_{0}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}\right]\to 0\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{1em}{0ex}}\mu \to \infty .$

Consequently the operator $\mu I-\mathcal{𝒜}$ is onto for sufficiety large $\mu >0$.

To prove that also the condition (ii) of Theorem 3.2.1 is fulﬁlled we consider an equivalent scalar product in $\text{H}$,

${〈\left[\begin{array}{c}\hfill {v}_{1}\hfill \\ \hfill {\psi }_{1}\hfill \end{array}\right],\left[\begin{array}{c}\hfill {v}_{2}\hfill \\ \hfill {\psi }_{2}\hfill \end{array}\right]〉}_{e}:={v}_{1}^{T}{v}_{2}+{\int }_{-r}^{0}{\psi }_{1}^{T}\left(\theta \right)\left(I-\frac{\theta }{r}{A}_{0}^{T}{A}_{0}\right){\psi }_{2}\left(\theta \right)d\theta .$

Then

$2{〈\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right],\mathcal{𝒜}\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]〉}_{e}={v}^{T}\left({A}_{1}+{A}_{1}^{T}\right)v+{v}^{T}\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)+{\psi }^{T}\left(-r\right){\left({A}_{1}{A}_{0}+{A}_{2}\right)}^{T}v$

$+{\int }_{-r}^{0}\frac{d}{d\theta }\left[{\psi }^{T}\left(\theta \right)\left(I-\frac{\theta }{r}{A}_{0}^{T}{A}_{0}\right)\psi \left(\theta \right)\right]d\theta +\frac{1}{r}{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){A}_{0}^{T}{A}_{0}\psi \left(\theta \right)d\theta =$

$=\left[\begin{array}{cc}\hfill {v}^{T}\hfill & \hfill {\psi }^{T}\left(-r\right)\hfill \end{array}\right]\left[\begin{array}{cc}\hfill {A}_{1}+{A}_{1}^{T}+I\hfill & \hfill {A}_{1}{A}_{0}+{A}_{2}+{A}_{0}\hfill \\ \hfill {\left({A}_{1}{A}_{0}+{A}_{2}+{A}_{0}\right)}^{T}\hfill & \hfill -I\hfill \end{array}\right]\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \left(-r\right)\hfill \end{array}\right]+$

$+\frac{1}{r}{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){A}_{0}^{T}{A}_{0}\psi \left(\theta \right)d\theta \le$

$\le {∥v∥}_{{\mathbb{ℝ}}^{n}}^{2}{∥{A}_{1}+{A}_{1}^{T}+I∥}_{L\left({\mathbb{ℝ}}^{n}\right)}+2{∥v∥}_{{\mathbb{ℝ}}^{n}}{∥\psi \left(-r\right)∥}_{{\mathbb{ℝ}}^{n}}{∥{A}_{1}{A}_{0}+{A}_{2}+{A}_{0}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}-$

$-{∥\psi \left(-r\right)∥}_{{\mathbb{ℝ}}^{n}}^{2}+\frac{1}{r}{∥{A}_{0}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}^{2}{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\left(I-\frac{\theta }{r}{A}_{0}^{T}{A}_{0}\right)\psi \left(\theta \right)d\theta \le$

$max\left\{\frac{1}{r}{∥{A}_{0}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}^{2},{∥{A}_{1}+{A}_{1}^{T}+I∥}_{L\left({\mathbb{ℝ}}^{n}\right)}+4{∥{A}_{1}{A}_{0}+{A}_{2}+{A}_{0}∥}_{L\left({\mathbb{ℝ}}^{n}\right)}^{2}\right\}{∥\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]∥}_{e}^{2}.$

The semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is EXS iff

 $\left|\sigma \left({A}_{0}\right)\right|<1$ (3.18)

i.e., the spectrum of ${A}_{0}$ is in an open unit circle and all roots of the characteristic quasipolynomial

 $\lambda ↦det\left[\lambda I-\lambda {e}^{-r\lambda }{A}_{0}-{A}_{1}-{e}^{-r\lambda }{A}_{2}\right]$ (3.19)

have negative real parts, see [30, Lemma 6.2.11, p. 151] for a proof. In what follows, we assume that (3.18) and (3.19) hold.

A linear observation operator $\mathcal{𝒞}\in L\left({D}_{\mathcal{𝒜}},{\mathbb{ℝ}}^{2n}\right)$ ($\text{Y}={\mathbb{ℝ}}^{2n}$),

$\mathcal{𝒞}\left[\begin{array}{c}v\hfill \\ \psi \hfill \end{array}\right]={\left[\begin{array}{cc}P\hfill & Q\hfill \\ {Q}^{T}\hfill & R\hfill \end{array}\right]}^{\frac{1}{2}}\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \left(-r\right)\hfill \end{array}\right]$

corresponds to the integrand in (3.16). Since the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is EXS we have

${\int }_{0}^{\infty }{∥z\left(t-r\right)∥}_{{\mathbb{ℝ}}^{n}}^{2}dt=\sum _{k=0}^{\infty }{\int }_{kr}^{\left(k+1\right)r}{∥z\left(t-r\right)∥}_{{\mathbb{ℝ}}^{n}}^{2}dt=\sum _{k=0}^{\infty }{\int }_{-r}^{0}{∥z\left(kr+\theta \right)∥}_{{\mathbb{ℝ}}^{n}}^{2}d\theta =$

$=\sum _{k=0}^{\infty }{\int }_{-r}^{0}{∥{z}_{kr}\left(\theta \right)∥}_{{\mathbb{ℝ}}^{n}}^{2}d\theta \le {M}^{2}{∥{x}_{0}∥}_{\text{H}}^{2}\sum _{k=0}^{\infty }{e}^{-2\mu kr}={M}^{2}{∥{x}_{0}∥}_{\text{H}}^{2}\frac{1}{1-{e}^{-2\mu r}}\phantom{\rule{1em}{0ex}}\forall {x}_{0}\in \text{H}.$

Employing the Rayleigh inequality we get

${∥\mathcal{𝒞}S\left(\cdot \right)\left[\begin{array}{c}{v}_{0}\hfill \\ \phi \hfill \end{array}\right]∥}_{{\text{L}}^{2}\left(0,\infty ;{\mathbb{ℝ}}^{2n}\right)}^{2}\le$

$\le {\lambda }_{max}\left(\left[\begin{array}{cc}P\hfill & Q\hfill \\ {Q}^{T}\hfill & R\hfill \end{array}\right]\right)\left[\frac{1}{2\mu }+\frac{1}{1-{e}^{-2\mu r}}\right]{M}^{2}{∥{x}_{0}∥}_{\text{H}}^{2}\phantom{\rule{2em}{0ex}}\forall {x}_{0}\in D\left(\mathcal{𝒜}\right)$

and thus (2.21) holds, i.e., $\mathcal{𝒞}$ is admissible. It follows from Theorems 2.4.1, 2.4.2, and (2.24) that

$J\left({x}_{0}\right)={〈{x}_{0},\mathcal{ℋ}{x}_{0}〉}_{\text{H}}\phantom{\rule{2em}{0ex}}\forall {x}_{0}\in \text{H}$

where $\mathcal{ℋ}$ is a unique bounded self–adjoint nonnegative solution to the Lyapunov operator equation (2.22) which reduces now to

${〈\mathcal{𝒜}x,\mathcal{ℋ}x〉}_{\text{H}}+{〈x,\mathcal{ℋ}\mathcal{𝒜}x〉}_{\text{H}}=$

 $=-\left[{v}^{T},{\psi }^{T}\left(-r\right)\right]\left[\begin{array}{cc}P\hfill & Q\hfill \\ {Q}^{T}\hfill & R\hfill \end{array}\right]\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \left(-r\right)\hfill \end{array}\right]\phantom{\rule{2em}{0ex}}\forall x=\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]\in D\left(\mathcal{𝒜}\right)$ (3.20)

A solution of (3.20) will be sought in the form

 $\mathcal{ℋ}\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]=\left[\begin{array}{c}\hfill \alpha v+{\int }_{-r}^{0}\beta \left(\theta \right)\psi \left(\theta \right)d\theta \hfill \\ \hfill {\beta }^{T}\left(\cdot \right)v+{\int }_{-r}^{0}\delta \left(\cdot ,\sigma \right)\psi \left(\sigma \right)d\sigma +\gamma \psi \hfill \end{array}\right]$ (3.21)

with $\alpha ,\gamma \in L\left({\mathbb{ℝ}}^{n}\right)$, $\alpha ={\alpha }^{T}$, $\gamma ={\gamma }^{T}$,

 $\delta \left(\theta ,\sigma \right)=\left\{\begin{array}{cc}\Phi \left(\theta -\sigma \right),\phantom{\rule{0ex}{0ex}}\hfill & \phantom{\rule{0ex}{0ex}}\theta <\sigma \hfill \\ {\Phi }^{T}\left(\sigma -\theta \right),\phantom{\rule{0ex}{0ex}}\hfill & \phantom{\rule{0ex}{0ex}}\theta >\sigma \hfill \end{array}\right\}={\delta }^{T}\left(\sigma ,\theta \right)$ (3.22)

and $\Phi ,\beta \in {\text{C}}^{\infty }\left(\left[-r,0\right],L\left({\mathbb{ℝ}}^{n}\right)\right)$. The matrix kernel function (3.22) may have a discontinuity along the diagonal $\theta =\sigma$ of the square $\left[-r,0\right]×\left[-r,0\right]$, or equivalently, $\Phi \left(0\right)$ may not be a symmetric matrix.

Taking (3.17) and (3.21) into account in (3.20), and integrating by parts we get

${〈\left[\begin{array}{c}\hfill {A}_{1}v+\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)\hfill \\ \hfill {\psi }^{\prime }\hfill \end{array}\right],\left[\begin{array}{c}\hfill \alpha v+{\int }_{-r}^{0}\beta \left(\theta \right)\psi \left(\theta \right)d\theta \hfill \\ \hfill {\beta }^{T}\left(\cdot \right)v+{\int }_{-r}^{0}\delta \left(\cdot ,\sigma \right)\psi \left(\sigma \right)d\sigma +\gamma \psi \hfill \end{array}\right]〉}_{\text{H}}+$

$+{〈\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right],\left[\begin{array}{c}\hfill \alpha \left[{A}_{1}v+\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)\right]+{\int }_{-r}^{0}\beta \left(\theta \right){\psi }^{\prime }\left(\theta \right)d\theta \hfill \\ \hfill {\beta }^{T}\left(\cdot \right)\left[{A}_{1}v+\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)\right]+{\int }_{-r}^{0}\delta \left(\cdot ,\sigma \right){\psi }^{\prime }\left(\sigma \right)d\sigma +\gamma {\psi }^{\prime }\hfill \end{array}\right]〉}_{\text{H}}=$

$=\left[{v}^{T}{A}_{1}^{T}+{\psi }^{T}\left(-r\right)\left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\right]\left[\alpha v+{\int }_{-r}^{0}\beta \left(\theta \right)\psi \left(\theta \right)d\theta \right]+$

$+{\int }_{-r}^{0}{\left[{\psi }^{\prime }\left(\theta \right)\right]}^{T}{\beta }^{T}\left(\theta \right)vd\theta +{\int }_{-r}^{0}{\left[{\psi }^{\prime }\left(\theta \right)\right]}^{T}{\int }_{-r}^{0}\delta \left(\theta ,\sigma \right)\psi \left(\sigma \right)d\sigma d\theta +$

$+{\int }_{-r}^{0}{\left[{\psi }^{\prime }\left(\theta \right)\right]}^{T}\gamma \psi \left(\theta \right)d\theta +{v}^{T}\left[\alpha {A}_{1}v+\alpha \left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)\right]+{\int }_{-r}^{0}{v}^{T}\beta \left(\theta \right){\psi }^{\prime }\left(\theta \right)d\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\beta }^{T}\left(\theta \right)\left[{A}_{1}v+\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)\right]d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\int }_{-r}^{0}\delta \left(\theta ,\sigma \right){\psi }^{\prime }\left(\sigma \right)d\sigma d\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\gamma {\psi }^{\prime }\left(\theta \right)d\theta ={v}^{T}{A}_{1}^{T}\alpha v+{\psi }^{T}\left(-r\right)\left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\alpha v+$

$+{\int }_{-r}^{0}{v}^{T}{A}_{1}^{T}\beta \left(\theta \right)\psi \left(\theta \right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(-r\right)\left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\beta \left(\theta \right)\psi \left(\theta \right)d\theta +$

$+{\int }_{-r}^{0}\frac{d}{d\theta }\left[{\psi }^{T}\left(\theta \right){\beta }^{T}\left(\theta \right)\right]vd\theta -{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\frac{d{\beta }^{T}\left(\theta \right)}{d\theta }vd\theta +$

$+{\int }_{-r}^{0}{\left[{\psi }^{\prime }\left(\theta \right)\right]}^{T}{\int }_{-r}^{\theta }{\Phi }^{T}\left(\sigma -\theta \right)\psi \left(\sigma \right)d\sigma d\theta +{\int }_{-r}^{0}{\left[{\psi }^{\prime }\left(\theta \right)\right]}^{T}{\int }_{\theta }^{0}\Phi \left(\theta -\sigma \right)\psi \left(\sigma \right)d\sigma d\theta +$

$+{v}^{T}\alpha \left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)+{\int }_{-r}^{0}\frac{d}{d\theta }\left[{\psi }^{T}\left(\theta \right)\gamma \psi \left(\theta \right)\right]d\theta -{\int }_{-r}^{0}{v}^{T}\frac{d\beta \left(\theta \right)}{d\theta }\psi \left(\theta \right)d\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\beta }^{T}\left(\theta \right){A}_{1}vd\theta +{\int }_{-r}^{0}{v}^{T}\frac{d}{d\theta }\left[\beta \left(\theta \right)\psi \left(\theta \right)\right]d\theta +{v}^{T}\alpha {A}_{1}v+$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\beta }^{T}\left(\theta \right)\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\int }_{-r}^{\theta }{\Phi }^{T}\left(\sigma -\theta \right){\psi }^{\prime }\left(\sigma \right)d\sigma d\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\int }_{\theta }^{0}\Phi \left(\theta -\sigma \right){\psi }^{\prime }\left(\sigma \right)d\sigma d\theta ={v}^{T}{A}_{1}^{T}\alpha v+{\psi }^{T}\left(-r\right)\left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\alpha v+$

$+{\int }_{-r}^{0}{v}^{T}{A}_{1}^{T}\beta \left(\theta \right)\psi \left(\theta \right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(-r\right)\left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\beta \left(\theta \right)\psi \left(\theta \right)d\theta +$

$+\left[{v}^{T}+{\psi }^{T}\left(-r\right){A}_{0}^{T}\right]{\beta }^{T}\left(0\right)v-{\psi }^{T}\left(-r\right){\beta }^{T}\left(-r\right)v-{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\frac{d{\beta }^{T}\left(\theta \right)}{d\theta }vd\theta -$

$-{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\Phi }^{T}\left(0\right)\psi \left(\theta \right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\int }_{-r}^{\theta }{\left[{\Phi }^{\prime }\left(\sigma -\theta \right)\right]}^{T}\psi \left(\sigma \right)d\sigma d\theta +$

$+{\int }_{-r}^{0}{v}^{T}{\Phi }^{T}\left(\theta \right)\psi \left(\theta \right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(-r\right){A}_{0}^{T}{\Phi }^{T}\left(\theta \right)\psi \left(\theta \right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\Phi \left(0\right)\psi \left(\theta \right)d\theta -$

$-{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\int }_{\theta }^{0}{\Phi }^{\prime }\left(\theta -\sigma \right)\psi \left(\sigma \right)d\sigma d\theta -{\int }_{-r}^{0}{\psi }^{T}\left(-r\right)\Phi \left(-r-\theta \right)\psi \left(\theta \right)d\theta +$

$+\left[{v}^{T}+{\psi }^{T}\left(-r\right){A}_{0}^{T}\right]\gamma \left[v+{A}_{0}\psi \left(-r\right)\right]-{\psi }^{T}\left(-r\right)\gamma \psi \left(-r\right)+{v}^{T}\alpha {A}_{1}v+$

$+{v}^{T}\alpha \left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)+{v}^{T}\beta \left(0\right)\left[v+{A}_{0}\psi \left(-r\right)\right]-{v}^{T}\beta \left(-r\right)\psi \left(-r\right)-$

$-{\int }_{-r}^{0}{v}^{T}\frac{d\beta \left(\theta \right)}{d\theta }\psi \left(\theta \right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\beta }^{T}\left(\theta \right){A}_{1}vd\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right){\beta }^{T}\left(\theta \right)\left({A}_{1}{A}_{0}+{A}_{2}\right)\psi \left(-r\right)d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\left[{\Phi }^{T}\left(0\right)\psi \left(\theta \right)-{\Phi }^{T}\left(-r-\theta \right)\psi \left(-r\right)\right]d\theta -$

$-{\int }_{-r}^{0}{\int }_{-r}^{\theta }{\psi }^{T}\left(\theta \right){\left[{\Phi }^{\prime }\left(\sigma -\theta \right)\right]}^{T}\psi \left(\sigma \right)d\sigma d\theta +{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\Phi \left(\theta \right)vd\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\Phi \left(\theta \right){A}_{0}\psi \left(-r\right)d\theta -{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\Phi \left(0\right)\psi \left(\theta \right)d\theta +{\int }_{-r}^{0}{\int }_{\theta }^{0}{\psi }^{T}\left(\theta \right){\Phi }^{\prime }\left(\theta -\sigma \right)\psi \left(\sigma \right)d\sigma d\theta$

$={v}^{T}\left[{A}_{1}^{T}\alpha +\alpha {A}_{1}+{\beta }^{T}\left(0\right)+\beta \left(0\right)+\gamma \right]v+{v}^{T}\left[\gamma {A}_{0}+\alpha \left({A}_{1}{A}_{0}+{A}_{2}\right)+\beta \left(0\right){A}_{0}-\beta \left(-r\right)\right]\psi \left(-r\right)+$

$+{\psi }^{T}\left(-r\right)\left[{A}_{0}^{T}\gamma +\left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\alpha +{A}_{0}^{T}{\beta }^{T}\left(0\right)-{\beta }^{T}\left(-r\right)\right]v+{\psi }^{T}\left(-r\right)\left[{A}_{0}^{T}\gamma {A}_{0}-\gamma \right]\psi \left(-r\right)+$

$+{\int }_{-r}^{0}{v}^{T}\left[{A}_{1}^{T}\beta \left(\theta \right)-\frac{d\beta \left(\theta \right)}{d\theta }+{\Phi }^{T}\left(\theta \right)\right]\psi \left(\theta \right)d\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\left[-\frac{d{\beta }^{T}\left(\theta \right)}{d\theta }+{\beta }^{T}\left(\theta \right){A}_{1}+\Phi \left(\theta \right)\right]vd\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(-r\right)\left[\left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\beta \left(\theta \right)-\Phi \left(-r-\theta \right)+{A}_{0}^{T}{\Phi }^{T}\left(\theta \right)\right]\psi \left(\theta \right)d\theta +$

$+{\int }_{-r}^{0}{\psi }^{T}\left(\theta \right)\left[{\beta }^{T}\left(\theta \right)\left({A}_{1}{A}_{0}+{A}_{2}\right)+\Phi \left(\theta \right){A}_{0}-{\Phi }^{T}\left(-r-\theta \right)\right]\psi \left(-r\right)d\theta =$

$=-\left[\begin{array}{cc}\hfill {v}^{T}\hfill & \hfill {\psi }^{T}\left(-r\right)\hfill \end{array}\right]\left[\begin{array}{cc}\hfill P\hfill & \hfill Q\hfill \\ \hfill {Q}^{T}\hfill & \hfill R\hfill \end{array}\right]\left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \left(-r\right)\hfill \end{array}\right]\phantom{\rule{2em}{0ex}}\forall \left[\begin{array}{c}\hfill v\hfill \\ \hfill \psi \hfill \end{array}\right]\in D\left(\mathcal{𝒜}\right).$

Hence we come to a system of equation determining $\alpha$, $\beta$, $\gamma$ and $\delta$,

 $\left\{\begin{array}{ccc}\hfill {A}_{1}^{T}\alpha +\alpha {A}_{1}+{\beta }^{T}\left(0\right)+\beta \left(0\right)+\gamma & \hfill =\hfill & \hfill -P\\ \hfill \gamma {A}_{0}+\alpha \left({A}_{1}{A}_{0}+{A}_{2}\right)+\beta \left(0\right){A}_{0}-\beta \left(-r\right)& \hfill =\hfill & \hfill -Q\\ \hfill {A}_{0}^{T}\gamma {A}_{0}-\gamma & \hfill =\hfill & \hfill -R\\ \hfill {A}_{1}^{T}\beta \left(\theta \right)-\frac{d\beta \left(\theta \right)}{d\theta }+{\Phi }^{T}\left(\theta \right)& \hfill =\hfill & \hfill 0\\ \hfill \left({A}_{2}^{T}+{A}_{0}^{T}{A}_{1}^{T}\right)\beta \left(\theta \right)-\Phi \left(-r-\theta \right)+{A}_{0}^{T}{\Phi }^{T}\left(\theta \right)& \hfill =\hfill & \hfill 0\end{array}\right\}$ (3.23)

By elimination of $\Phi$ we reduce (3.23) to the discrete Lyapunov matrix equation

 ${A}_{0}^{T}\gamma {A}_{0}-\gamma =-R$ (3.24)

and the boundary–value problem

 $\left\{\begin{array}{c}\frac{d}{d\theta }\left[\beta \left(\theta \right)+{\beta }^{T}\left(-r-\theta \right){A}_{0}\right]={A}_{1}^{T}\beta \left(\theta \right)+{\beta }^{T}\left(-r-\theta \right){A}_{2}\hfill \\ {A}_{1}^{T}\alpha +\alpha {A}_{1}+{\beta }^{T}\left(0\right)+\beta \left(0\right)+\gamma =-P\hfill \\ \gamma {A}_{0}+\alpha \left({A}_{1}{A}_{0}+{A}_{2}\right)+\beta \left(0\right){A}_{0}-\beta \left(-r\right)=-Q\hfill \end{array}\right\}$ (3.25)

Furthermore, we get also

 $\Phi \left(\theta \right)=\frac{d{\beta }^{T}\left(\theta \right)}{d\theta }-{\beta }^{T}\left(\theta \right){A}_{1}={A}_{2}^{T}\beta \left(-r-\theta \right)-{A}_{0}^{T}\frac{d\beta \left(-r-\theta \right)}{d\theta }$ (3.26)

Remark 3.2.1. Castelan and Infante ,  have derived (3.25) in the case ${A}_{0}=0$, i.e., for retarded systems and a much more complicated version of (3.25) for neutral systems provided that ${\text{W}}^{1,2}\left(-r,0;{\mathbb{ℝ}}^{n}\right)$ was chosen as a state space.

A special technique has been developed in  for the analysis of their version of the problem (3.25). In what follows we adapt that technique to solve (3.25). By substituting

 $\vartheta \left(\theta \right)={\beta }^{T}\left(-r-\theta \right),\phantom{\rule{2em}{0ex}}-r\le \theta \le 0$ (3.27)

one can reduce the ﬁrst equation of (3.25) to the system

 $\left\{\begin{array}{ccc}\hfill \frac{d}{d\theta }\left[\beta \left(\theta \right)+\vartheta \left(\theta \right){A}_{0}\right]& \hfill =\hfill & \hfill {A}_{1}^{T}\beta \left(\theta \right)+\vartheta \left(\theta \right){A}_{2}\\ \hfill \frac{d}{d\theta }\left[{A}_{0}^{T}\beta \left(\theta \right)+\vartheta \left(\theta \right)\right]& \hfill =\hfill & \hfill -{A}_{2}^{T}\beta \left(\theta \right)-\vartheta \left(\theta \right){A}_{1}\end{array}\right\}$ (3.28)

In turn, (3.28) is equivalent to a linear autonomous system in the space ${\mathbb{ℝ}}^{{n}^{2}}$ which can be seen by applying the Kronecker product of matrices ([57, Section 8.4]). This yields

$\frac{d}{d\theta }\left[\begin{array}{cc}\hfill I\otimes I& I\otimes {A}_{0}^{T}\hfill \\ \hfill {A}_{0}^{T}\otimes I& I\otimes I\hfill \end{array}\right]\left[\begin{array}{c}\hfill col\phantom{\rule{0ex}{0ex}}\beta \hfill \\ \hfill col\phantom{\rule{0ex}{0ex}}\vartheta \hfill \end{array}\right]=\left[\begin{array}{cc}\hfill {A}_{1}^{T}\otimes I& \hfill I\otimes {A}_{2}^{T}\\ \hfill -{A}_{2}^{T}\otimes I& \hfill -I\otimes {A}_{1}^{T}\end{array}\right]\left[\begin{array}{c}\hfill col\phantom{\rule{0ex}{0ex}}\beta \hfill \\ \hfill col\phantom{\rule{0ex}{0ex}}\vartheta \hfill \end{array}\right],$

where $col\phantom{\rule{0ex}{0ex}}\beta$, $col\phantom{\rule{0ex}{0ex}}\vartheta$ denote ${n}^{2}$–dimensional vectors having rows composed of the rows of matrices $\beta$ and $\vartheta$, respectively. By the Schur lemma and (3.18) we have

$det\left[\begin{array}{cc}\hfill I\otimes I& I\otimes {A}_{0}^{T}\hfill \\ \hfill {A}_{0}^{T}\otimes I& I\otimes I\hfill \end{array}\right]=det\left(I\otimes I-{A}_{0}^{T}\otimes {A}_{0}^{T}\right)\ne 0.$

Hence

$\frac{d}{d\theta }\left[\begin{array}{c}\hfill col\phantom{\rule{0ex}{0ex}}\beta \hfill \\ \hfill col\phantom{\rule{0ex}{0ex}}\vartheta \hfill \end{array}\right]={\left[\begin{array}{cc}\hfill I\otimes I& I\otimes {A}_{0}^{T}\hfill \\ \hfill {A}_{0}^{T}\otimes I& I\otimes I\hfill \end{array}\right]}^{-1}\left[\begin{array}{cc}\hfill {A}_{1}^{T}\otimes I& \hfill I\otimes {A}_{2}^{T}\\ \hfill -{A}_{2}^{T}\otimes I& \hfill -I\otimes {A}_{1}^{T}\end{array}\right]\left[\begin{array}{c}\hfill col\phantom{\rule{0ex}{0ex}}\beta \hfill \\ \hfill col\phantom{\rule{0ex}{0ex}}\vartheta \hfill \end{array}\right].$

Employing again the Schur lemma and some properties of the Kronecker product we ﬁnd the characteristic polynomial of the above system,

 $det\left[\left(\lambda I-{A}_{1}^{T}\right)\otimes \left(\lambda I+{A}_{1}^{T}\right)+\left({A}_{2}^{T}+\lambda {A}_{0}^{T}\right)\otimes \left({A}_{2}^{T}-\lambda {A}_{0}^{T}\right)\right]$ (3.29)

Thus

 ${e}^{\lambda \theta }\left[\begin{array}{c}\hfill L\hfill \\ \hfill M\hfill \end{array}\right]$ (3.30)

is an eigensolution of (3.28) where $\lambda$ is a root of (3.29), and matrices $L$, $M\in L\left({\mathbb{ℂ}}^{{n}^{2}}\right)$ satisfy the system

 $\left\{\begin{array}{ccc}\hfill \lambda L+\lambda M{A}_{0}& \hfill =\hfill & {A}_{1}^{T}L+M{A}_{2}\hfill \\ \hfill \lambda {A}_{0}^{T}L+\lambda M& \hfill =\hfill & -{A}_{2}^{T}L-M{A}_{1}\hfill \end{array}\right\}$ (3.31)

By multiplying the equations of (3.31) by $\left(-1\right)$, transposing and reordering them, one can see that if (3.30) is an eigensolution then ${e}^{-\lambda \theta }\left[\begin{array}{c}{M}^{T}\hfill \\ {L}^{T}\hfill \end{array}\right]$is an eigensolution too.

Assume from now that all eigenvalues of (3.28) have linear elementary divisors. Then the corresponding eigenvectors form a basis in ${\mathbb{ℂ}}^{{n}^{2}}$ and the general solution of (3.28) is

$\left[\begin{array}{c}\hfill \beta \left(\theta \right)\hfill \\ \hfill \vartheta \left(\theta \right)\hfill \end{array}\right]=\sum _{i=1}^{{n}^{2}}\left\{{\kappa }_{i}{e}^{{\lambda }_{i}\theta }\left[\begin{array}{c}{L}_{i}\hfill \\ {M}_{i}\hfill \end{array}\right]+{\mu }_{i}{e}^{-{\lambda }_{i}\theta }\left[\begin{array}{c}\hfill {M}_{i}^{T}\hfill \\ \hfill {L}_{i}^{T}\hfill \end{array}\right]\right\}.$

It is easy to see that this solution satisﬁes the functional equation (3.27) if and only if ${\mu }_{i}={\kappa }_{i}{e}^{-{\lambda }_{i}r}$ and ﬁnally

 $\beta \left(\theta \right)=\sum _{i=1}^{{n}^{2}}{\kappa }_{i}\left[{e}^{{\lambda }_{i}\theta }{L}_{i}+{e}^{-{\lambda }_{i}\left(r+\theta \right)}{M}_{i}^{T}\right]$ (3.32)

is a general solution of the ﬁrst equation of (3.25). Putting (3.32) into the second and third equation of (3.25) yields

$\begin{array}{ccc}\hfill \gamma +{A}_{1}^{T}\alpha +\alpha {A}_{1}+\sum _{i=1}^{{n}^{2}}{\kappa }_{i}\left[{L}_{i}+{L}_{i}^{T}+{e}^{-{\lambda }_{i}r}\left({M}_{i}+{M}_{i}^{T}\right)\right]& \hfill =\hfill & -P\hfill \\ \hfill \gamma {A}_{0}+\alpha \left({A}_{1}{A}_{0}+{A}_{2}\right)+\sum _{i=1}^{{n}^{2}}{\kappa }_{i}\left[{e}^{-{\lambda }_{i}r}\left({M}_{i}^{T}{A}_{0}-{L}_{i}\right)+\left({L}_{i}{A}_{0}-{M}_{i}^{T}\right)\right]& \hfill =\hfill & -Q.\hfill \end{array}$

A next application of the Kronecker product of matrices enables us to represent the last equations as

 (3.33)

The matrix of the system (3.33) is nonsingular. Indeed, if this is not the case, then taking $P=Q=R=0$ (in virtue of (3.24) and (3.18) we also have $\gamma =0$) and making use of formulae (3.33), (3.32), (3.26), (3.24) and (3.21) we can generate matrices $\alpha$, $\beta \left(\theta \right)$, $\delta \left(\theta ,\sigma \right)$, $\gamma$ and thus a nonzero operator $\mathcal{ℋ}$ being a solution to the Lyapunov operator equation (3.20). However, this contradicts the uniqueness of the null solution for $\mathcal{𝒞}=0$ which, in turn, follows from (3.18), (3.19) and Theorem 2.4.2. Finally, (3.33) has a unique solution which means that formulae: (3.33), (3.32), (3.26), (3.24) and (3.21) determine matrices $\alpha$, $\beta \left(\theta \right)$, $\delta \left(\theta ,\sigma \right)$, $\gamma$ and thus an operator $\mathcal{ℋ}$ being the unique solution of the Lyapunov operator equation (3.20).

The assumption that all eigenvalues of (3.28) are single is not essential for validity of the above derivation and (3.32) can be appropriately modiﬁed if there are nonlinear elementary divisors.

Let us indicate two possible simpliﬁcations of the performance index evaluation which can arise in practical applications. The ﬁrst is symmetry of matrices $\alpha$, $\gamma$, $P$ which causes that (3.33) contains $\frac{n\left(n-1\right)}{2}$ redundant equations. The second is that for a large variety of initial conditions the evaluation of the performance index does not require the knowledge of all entries of $\alpha$, $\beta \left(\theta \right)$, $\delta \left(\theta ,\sigma \right)$, $\gamma$ (e.g. for ${x}_{0}=\left[\begin{array}{c}\hfill {v}_{0}\hfill \\ \hfill 0\hfill \end{array}\right]$it suffices to determine only the matrix $\alpha$).

The Kronecker product of matrices can also be applied (see ) to derive the frequency–domain method of evaluation the performance index

$J\left(\left[\begin{array}{c}\hfill {v}_{0}\hfill \\ \hfill \phi \hfill \end{array}\right]\right)={\int }_{0}^{\infty }{z}^{T}\left(t\right){Q}_{0}z\left(t\right)dt.$