Studying the Dynamic Properties of a Distributed Thermomechanical System and Stability Conditions for Its Control System

Proof of Theorem 1. Let us perform the Laplace transform of Eq. (5.1) with respect to the spatial coordinate \(x\in \left [0, l\right ] \) with allowance for the first of the boundary conditions in (5.2) (see [11, item 80, formula (7)]),

where \(\overline {\overline {f}}\) is the Laplace transform of the function \( \overline {f}\) (\(\overline {f}(x)(p)\in \mathbf {C}\)) with respect to \(x \) and

This implies the following expression for \(\overline {\overline {\varphi }}\left (q\right )\):

where \(b=\frac {\beta }{c^2} \).

We obtain the expression for \(\overline {\overline {\theta }}\left (q\right ) \) by performing the Laplace transform of the expression (3.2) for \(\overline {\theta } \) with respect to the coordinate \(x \). By virtue of formulas (4) in [11, item 80], the transformations (with respect to \(x \)) of the functions \(\cosh \left (\frac {l-x}{l}\zeta \right )\ \) and \(\sinh \left (\frac {l-x}{l}\zeta \right ) \) occurring in (3.2) have the respective form

Therefore, the expression for \(\overline {\overline {\theta }}\left (q\right ) \) has the form

where \(A_c\left (\zeta \right )={\cosh \zeta +\frac {r}{\zeta } }\;{\sinh \zeta } \) and \(A_s\left (\zeta \right )=\frac {r}{l}\;{\cosh \zeta +\frac {\zeta }{l}}\;{\sinh \zeta } \).

After substituting the expression (A.1.3) for \(\overline {\overline {\theta }}\left (q\right ) \) into (A.1.2), we obtain the rational fraction

where \(\gamma (p)=ap-c^2\).

Using this relation and the following representation for \(\overline {\theta }\left (0\right )\) implied by (3.2): \(\overline {\theta }\left (0\right ) =\frac {A_c\left (\zeta \right )}{D_{\mathrm{T}}\left (\zeta \right )}\overline {u} \) (see the explanations to (A.1.3)), we obtain the ultimate expression for \(\overline {\overline {\varphi }}\left (q\right )\),

Based on this, we pass to the originals with respect to \(x \),

Further, differentiating (A.1.5) with respect to \(x \), we conclude that the second of the boundary conditions in (5.2) takes the form

where \(\sigma (p)=\sinh \left (\frac {p}{c}l\right )\).

Eliminating \(\overline {\varphi }\left (0\right ) \) from (A.1.5) and (A.1.6) and using the expressions for the functions \(A_c \) and \(A_{s } \) (see the explanations to (A.1.3)), we obtain the ultimate expression for \(\overline {\varphi }\left (x\right )\) in the form (5.3).

Proof of Theorem 2 .

1. Let us represent the function \(\Psi \) (see (5.4)) in the form of the sum \({\Psi }_1+{\Psi }_2 \), where

To study the dynamic properties of the operators \(V_{{\Psi }_j} \) (\(j=1,2 \)) with the transfer functions \({\Psi }_j \), we use the Cauchy theorem [11, item 71].

To apply this theorem, we need to construct a regular system of contours \(\left \{{\mathbf {G}_n,}\; n\ge n_0\right \} \) and check whether the values of the functions \( {\Psi }_j\) (\(j=1, 2 \)) tend to zero on these contours as \(n\to \infty \).

2. First, we introduce the domain \(\mathbf {H}_0=\left \{p\in \mathbf {C}:\ \left |p\right |>\left |p_{\mathrm{oc}}\right |\right \} \) (where \(p_{\mathrm{oc}} \) is as described in item 5), and in this domain we construct a system of contours in the form of rectangles with sides \(\mathbf {G}_{\pm \mathrm{g}n}=\left \{p\in \mathbf {C}:\left |\mathrm {Re}p\right |\le H_n, \mathrm {Im}p=\pm L_n\right \}\) and \(\mathbf {G}_{\pm \mathrm{b}\boldsymbol {n}}=\left \{p\in \mathbf {C}:\mathrm {Re}p={\pm H}_n,\left |\mathrm {Im}p\right |\le L_{\boldsymbol {n}}\right \} \), where the numbers \(H_n \) and \(L_{\boldsymbol {n}} \) are to be chosen below.

In this case, for the system of contours \(\left \{\mathbf {G}_{\boldsymbol {n}},n\ge n_0\right \}\) to be in the domain \( \mathbf {H}_0\), the number \(n_0 \) must satisfy the condition \(\min \left (H_{n_0},L_{n_0}\right )>\left |p_{\mathrm{oc}}\right | \).

3. We define the quantity \(H_n\) so that the side \(\mathbf {G}_{-\mathrm{b}\boldsymbol {n}}\) passes through the point \(p_{en}=-a{\left (\frac {\tau _{en}}{l}\right )}^2 \) of the plane \(\mathbf {C} \); here \(\tau _{en} \) is the point of maximum of the function \(D_{\mathrm {T}}\left (i\tau \right )=\propto \left [2{\cos \tau }+\left (\frac {r}{\tau }-\frac {\tau }{r}\right ){\sin \tau }\right ] \) (see the explanations to (3.2)).

Since \(\frac {\partial }{\partial t}\left (D_{\mathrm{T}}\left (i\tau \right )\right )=\propto \left [\left (\frac {r}{\tau }-\frac {\tau }{r}\right ){\cos \tau }-\left (2+\frac {r}{\tau ^2}+\frac {1}{r}\right ){\sin \tau }\right ]\), we see that the value of \(\tau _{en} \) is determined as the solution of the equation

where \(f_D\left (\tau \right )=\frac {r^2-\tau ^2}{r^2+\left (2r+1\right )\tau ^2}\). The values of \(\sin \tau _{en} \) and \({{\cos \tau }}_{en} \) have different signs on the solutions of this equation, which alternate with \(n\) increasing; as \(n\to \infty \), we have \(\sin \tau _{en} \to 1 \) and \(\cos \tau _{en} \to 0 \).

Thus, the quantity \(H_n\) is determined as \({a\left (\frac {\tau _{en}}{l}\right )}^2\). Since the number \(\tau _{en} \) is within the interval \(\left (\tau _n,\tau _{n+1}\right )\) and \(\tau _n\in \left (n, n+1\right )\pi \), we see that the quantity \(H_n \) grows with increasing \(n \) and has the growth order of \({n}^2 \).

4. The value of \(L_n\) is chosen so that the side \(\mathbf {G}_{\pm \mathrm{g}\boldsymbol {n}}\) does not pass through the zeros of the function \(\sigma \) (see the explanations to (5.3)). Namely, \(L_n \) is determined as \(\frac {c}{l}\pi \left (n^2+\frac {1}{2}\right )\). (For the system \(\left \{\mathbf {G}_{n},n\ge n_0\right \}\) of contours to be regular, the values of \(L_n\) must have the same growth order as \(H_n\).)

5. Let us estimate the values of functions occurring in the expressions (A.2.1) and (A.2.2) for \({\Psi }_j\), \(j=1,2 \), on the sides of the contour \(\mathbf {G}_{\boldsymbol {n}} \) using the properties of hyperbolic functions of complex variable. In this case, by virtue of the complex conjugacy of numbers on the sides \(\mathbf {G}_{+\mathrm{g}n}\) and \(\mathbf {G}_{-\mathrm{g}\boldsymbol {n}}\), it suffices to restrict ourselves to the upper half of the plane \(\mathbf {C}\).

(a) By virtue of the choice of \(L_n\) on the side \(\mathbf {G}_{+\mathrm{g}\boldsymbol {n}}\), the value of the function \( \sigma \) (see the explanations to (5.3)) is equal to \(i\thinspace {\cosh \thinspace \left (\frac {l}{c}\mathrm {Re}p\right )}\), and the values of the function \(\left |{\cosh \left (\frac {\xi }{c}p\right )}\right | \) (where \(\xi \in \left [0, l\right ] \)) do not exceed \(\cosh \left (\frac {l}{c}\mathrm {Re}p\right )\). Therefore, the values of the ratio \(\left |\frac {{\cosh \left (\left (\xi /c\right )p\right ) }}{\sigma (p)}\right |\) are at most 1.

(b) Since the absolute value of the complex number \(\zeta (p) \) (see the explanations to (3.2)) is equal to \(\sqrt {\frac {\left |p\right |}{a}}l\) and its argument is half the argument of the number \(p \), we have \({\arg p }\in \left [\frac {\pi }{2}-\delta _n,\frac {\pi }{2}+\delta _n\right ] \) on the side \(\mathbf {G}_{+\mathrm{g}n} \), where \(\delta _n={\arctan \frac {H_n}{L_n} }\), and consequently, \({\arg \zeta (p)\in \left [\frac {\pi }{4}-\frac {\delta _n}{2}, \frac {\pi }{4}+\frac {\delta _n}{2}\right ] }\) and

Here we have denoted \(M_n=\sqrt {H^2_n+L^2_n} \) and used the relation \(\sqrt {M_n-L_n}=\frac {H_n}{\sqrt {M_n+L_n}}\).

Since \(\left |\zeta (p)\right |\ge \sqrt {\frac {L_n}{a}}l \) on \(\mathbf {G}_{+\mathrm{g}n} \), it follows that the values of the function \(\mathrm {Re} \thinspace \zeta (p) \) increase as \(n\to \infty \) with growth order of at least \(n \).

(c) We introduce the notation \(z_1\left (\zeta \right )=\frac {\zeta }{r}+\frac {r}{\zeta }\) and write down the expanded expression for \({D}_{\mathrm{T}}\left (\zeta \right )\),

This implies the representation of the function \({\left |D_{\mathrm {T}}\left (\zeta \right )\right |}^2\) in the form of the sum \(\sum ^2_{j=1}{F_{Dj}\left (\zeta \right )}\), where

The function \(F_{D1}(\zeta )\) is estimated from below by the function

To estimate the values of the function \(F_{D2}(\zeta ) \) for large \(n \), note that the values of the function \(z_1(\zeta ) \) approach the values of the function \(\frac {\zeta }{r}\) as \(n \) grows. The lower bound for the growth of the function \(\mathrm {Re}\thinspace \zeta \) has been obtained in (A.2.4). At the same time, the estimate of growth for the function \( \mathrm {Im}\thinspace \zeta \) can be obtained by analogy with (A.2.4),

It can be seen from this estimate that for sufficiently large \(n \) the values of the function \(F_{D2}(\zeta ) \) become positive, and therefore, the function \(F_D\left (\zeta \right )\) becomes a lower bound for the whole function \( {\left |D_{\mathrm {T}}(\zeta )\right |}^2\).

(d) The values of the modulus of the function \(A_c\left (\zeta \right ) \) (see the explanations to (5.3)) occurring in the expression (A.2.1) for the function \({\Psi }_1(\zeta ) \) are estimated from above as

and therefore, \(\left |\frac {A_c\left (\zeta \right )}{D_{\mathrm{T}}\left (\zeta \right )}\right |\le \frac {1}{\propto \left |z_1\left (\zeta \right )\right |} \left (1+\frac {r}{\left |\zeta \right |}\right ) {\coth \mathrm {Re}\thinspace \zeta } \).

Thus, the ratio \(\frac {A_c\left (\zeta \right )}{D_{\mathrm {T}}\left (\zeta \right )} \) tends to zero as \(n\to \infty \).

It follows from items (a)–(d) that the values of the function \({\Psi }_1 \) on the side \(\mathbf {G}_{+\mathrm{g}\boldsymbol {n}}\) tend to zero as \(n\to \infty \).

(e) The absolute value of the function \({z}_2(p)=\frac {ar}{l}\cosh \left (\frac {l-x}{l} \zeta (p)\right )+\sqrt {ap}\sinh \left (\frac {l-x}{l}\zeta (p)\right )\) occurring in the expression (A.2.2) for the function \({\Psi }_2 \) can be estimated from above by the function \(\left (\frac {ar}{l}+\sqrt {a\left |p\right |}\right ) \cosh \left (\mathrm {Re}\thinspace \zeta (p)\right )\). The ratio of the function \(\left |z_2\right |\) to \(F_D(\zeta ) \) does not exceed the quantity \(\frac {ar/l+\sqrt {a\left |p\right |}}{\propto \left |z_1\left (\zeta (p)\right )\right |} \coth \thinspace (\mathrm {Re}\thinspace \zeta (p))\); i.e., it remains bounded as \(n\to \infty \). Since the expression (A.2.2) for \({\Psi }_2 \) contains the function \(\gamma \) with modulus that grows indefinitely as \(n\to \infty \) (see the explanations to (5.3)), we conclude that the values of the function \({\Psi }_2 \) on the side \(\mathbf {G}_{+\mathrm{g}n} \) tend to zero as \(n\to \infty \).

6. Let us estimate the values of the functions occurring in the expressions for \({\Psi }_j \) (\(j=1, 2 \)) on the side \(\mathbf {G}_{+\mathrm{b}n} \).

(a) The values of the function \(\left |\sigma \right | \) on \(\mathbf {G}_{+\mathrm{b}n} \) are estimated from below by the quantity \(\left |\sinh \left (\frac {l}{c}\mathrm {Re}\thinspace p\right )\right | \), i.e., by the quantity \(\sinh \left (\frac {l}{c}H_n\right ) \), and the absolute values of the function \( \cosh \left (\frac {\xi }{c}p\right )\) (where \(\xi \in \left [0,l \right ]\)) do not exceed the quantity \(\cosh \left (\frac {\xi }{c}\mathrm {Re}\thinspace p\right ) \), i.e., the quantity \(\cosh \left (\frac {l}{c}H_n\right )\). Therefore, the absolute values of the ratio \( \frac {\cosh \left (\left (\xi /c\right )p\right )}{\sigma (p)} \) on \(\mathbf {G}_{+\mathrm{b}n} \) do not exceed the quantity \(\coth \left (\frac {l}{c}H_{n_0}\right )\), where \(n_0 \) is as described in item 2.

(b)–(e) On the upper half of the side \( \mathbf {G}_{+\mathrm{b}n} \), we have \(\arg p\in \left [0, \frac {\pi }{2}-\delta _n\right ]\), where \(\delta _n \) is as described in item 5(b); therefore, \(\arg \zeta (p) \in \left [0,\frac {\pi }{4}-\frac {\delta _n}{2}\right ] \) and

Hence it follows that \(\mathrm {Re}\thinspace \zeta (p)\to \infty \) as \(n\to \infty \) with a growth order of at least \(n \). Therefore, the same conclusions as the ones in items 5(b)–5(e) are true regarding the functions \(\left |\frac {A_c}{D_{\mathrm{T}}}\left (\zeta \right )\right |\), \(\frac {z_2}{\gamma D_{\mathrm {T}}\circ \zeta }\), and \({\Psi }_j \) (\(j=1,2 \)) tending to zero as \(n\to \infty \).

7. Now let us estimate the values of the ratios contained in the expressions for the functions \({\Psi }_j\) (see (A.2.1), (A.2.2)) on the side \(\mathbf {G}_{-\mathrm{b}\boldsymbol {n}}\) of the contour \(\mathbf {G}_{\boldsymbol {n}}\). It suffices to perform this estimation for \( \mathrm {Im}\thinspace p\ge 0\).

(a) Repeating the reasoning in item 6(a), we conclude that the ratio \(\left |\frac {\cosh \left (\left (\xi /c\right )p\right )}{\sigma (p)}\right | \) (\(\xi \in \left [0,\ l\right ] \)) on the side \({\mathbf {G}}_{-\mathrm{b}n}\) can be estimated from above by the same quantity \(\coth \left (\frac {l}{c}H_{n_0}\right )\) as on the side \({\mathbf {G}}_{+\mathrm{b}n}\).

(b) On the upper half of the side \(\mathbf {G}_{-\mathrm{b}\boldsymbol {n}}\) we have \(\arg p \in \left [\frac {\pi }{2}+\delta _n,\pi \right ]\), and therefore, \({\arg \zeta (p) }\in \left [\frac {\pi }{4}+\frac {\delta _n}{2},\frac {\pi }{2}\right ] \). Introducing the angle \(\theta _n=\mathrm {arc}\cos \frac {H_n}{|p|} \), we have

At the same time, the quantity \(\mathrm {Im}\thinspace \zeta (p)\) is calculated as follows:

However, since \(H_n=a{\left (\frac {\tau _{en}}{l}\right )}^2 \) (see item 3 in the present appendix), we obtain

(c) For the quantity \({\sin }^2\mathrm {Im}\thinspace \zeta (p) \) (see (A.2.5)), we can write

It can be seen from (A.2.10) that under the condition \(\mathrm {Im}\;p<\sqrt {\frac {H_n}{\vartheta }} \), where \(\theta \) is a fixed constant of the dimension of time, the right-hand side of (A.2.10) tends to zero as \(n\to \infty \). However, since \(\left |\sin \tau _{en}\right |\to 1\) in this case (see item 3), it follows that \({\sin }^2\mathrm {Im}\;\zeta (p)\) tends to 1 as \(n\to \infty \) as well.

Taking into account this fact, we split the upper half of the side \(\mathbf {G}_{-\mathrm{b}\boldsymbol {n}}\) into the zone \(\boldsymbol {\Lambda }_n\), where \(\mathrm {Im}\thinspace p\in \left [\sqrt {\frac {H_n}{\theta }}, L_n\right ] \), and the zone \(\boldsymbol {\Lambda }_{0n} \), where \(\mathrm {Im} \thinspace p<\sqrt {\frac {H_n}{\theta }}\).

(d) In the zone \({\boldsymbol {\Lambda }}_n \), we have \({{\mathrm {arg}\; p \in \left [\frac {\pi }{2}+\delta _n,\pi -\theta _{0n}\right ]}} \), where \({\theta _{0n}=\arctan \sqrt {\frac {1}{\vartheta H_n}}}\); consequently, \({\arg \zeta (p) }\in \left [\frac {\pi }{4}+\frac {\delta _n}{2},\frac {\pi }{2}-\frac {\theta _{0n}}{2}\right ]\). Therefore,

where \(M_{0n}=\sqrt {H^2_n+H_n/\vartheta } \). Since \(\left |p\right |\ge M_{0n}\), we have this implies the lower boundedness of the values of the function \(\mathrm {Re}\;\zeta (p)\) as \(n \) grows indefinitely.

At the same time, the function \(\mathrm {Im}\thinspace \zeta (p) \) in the zone \(\boldsymbol {\Lambda }_n \) can be estimated from above as follows:

Now, estimating the values of the function \({\left |D_{\mathrm {T}}\left (\zeta \right )\right |}^2\) from below in the zone \( {\boldsymbol {\Lambda }}_n\) by the function

(see item 5c), we can see that for a sufficiently large \( n\) the function \({\left |D_{\mathrm {T}}\left (\zeta \right )\right |}^2\) is estimated from below by the function \(\frac {F_D\left (\zeta \right )}{2} \).

Therefore, based on the estimate (A.2.7), we obtain the following.

For a sufficiently large \(n\), one has an inequality similar to item 5(d),

By virtue of the lower boundedness of the function \(\mathrm {Re}\;\zeta \), this inequality implies that the ratio \(\left (\frac {A_c}{D_{\mathrm {T}}}\right ) \circ \zeta \) tends to zero as \( n\to \infty \).

In addition, by analogy with item 5(e), for sufficiently large \(n \) the modulus of the ratio \(\frac {z_{2 }(p)}{ D_{\mathrm{T}}\left (\zeta (p)\right )} \) is estimated from above by the quantity

By virtue of the lower boundedness of the function \(\mathrm {Re}\thinspace \zeta (p) \), the modulus of this ratio is bounded above as \(n \) grows indefinitely, and consequently, the ratio \(\frac {z_2}{\gamma D_{\mathrm {T}} \circ \zeta }\) tends to zero as \( n\to \infty \).

(e) In the zone \({\boldsymbol {\Lambda }}_{0n} \), we have

Since the function \({\sin }^2\thinspace \mathrm {Im}\thinspace \zeta \) tends to 1 as \(n\to \infty \), we can estimate the function \({\left |D_{\mathrm {T}}\left (\zeta \right )\right |}^2\rule {0pt}{4mm}\) from below in this zone by the function \({{F}_{D0}\left (\zeta \right )}{-2\alpha ^2\thinspace \mathrm {Im}\thinspace z_1\left (\zeta \right )\sin 2\thinspace \mathrm {Im}\thinspace \zeta }\), where

It follows from the form of the last two functions that, by analogy with the case of the zone \({\boldsymbol {\Lambda }}_n \) (see item 7(d)), for sufficiently large \(n \) the function \({\left |D_{\mathrm {T}}\left (\zeta \right )\right |}^2\) is estimated from below by the function \(\frac {F_{D0}\left (\zeta \right )}{2} \). Therefore, by analogy with item 7(d), for sufficiently large \( n\) the modulus of the ratio \(\frac {A_c\left (\zeta \right )}{D_{\mathrm{T}}\left (\zeta \right )} \) can be estimated from above by the modulus of the function \(2\frac {1+r/\zeta }{\alpha z_1\left (\zeta \right )}{\cosh \mathrm {Re}\thinspace \zeta } \) and hence tends to zero as \(n\to \infty \). At the same time, the modulus of the ratio \(\frac {z_2}{D_{\mathrm {T}}\left (\zeta \right )}\) can be estimated from above by the modulus of the function and hence remains bounded as \(n \) grows indefinitely. Consequently, the ratio \(\frac {z_2}{\gamma D_{\mathrm {T}}\circ \zeta } \) tends to zero as \( n\to \infty \).

8. It follows from items 2–7 that the values of the functions \({\Psi }_j \) (\(j=1, 2 \)) on the contours \({\mathbf {G}}_{n } \) (\(n\ge n_0 \)) tend to zero as \(n\to \infty \); this implies the possibility of applying the Cauchy theorem [11, item 71]. By this theorem, the function \({\Psi }_1\) is represented by the sum of the series composed of the principal parts of this function at its poles \(p_{\mathrm{oc}} \) and \(p_{n} \) (\(n\ge 0 \)) as well as at the zeros of the function \(\sigma \) (see Sec. 5). At the same time, the function \({\Psi }_2 \) is represented by the sum of the series composed of the principal parts of this function at its poles \({ p}_{\mathrm{oc}} \) and \(p_{n} \). Namely, since all these poles are simple, we have

where

Below, in Appendix 3, we provide the calculation of the values of the coefficients \(b_0\), \( b_{jn} \) (\(n\ge 0 \), \(j=1,2 \)), \(d_k \), and \(f_k \) (\(k\ge 1 \)) according to [11, item 23].

Adding (A.2.13) and (A.2.14), based on the results of calculations in Appendix 3, we obtain an expansion of the function \(\Psi \),

where

The expressions for the coefficients \(b_0 \) and \(b_{jn} \) are derived in Appendix 3, items 1 and 3, respectively; the expressions for the coefficients \(d_k \) and \(f_k \) are obtained in item 5 of the same appendix; the expression for \( \left (D_{\mathrm {T}}\right )^{\prime }\left (p_n\right ) \) is derived item 4 of the same appendix. Since the values of \({\left (D_{\mathrm{T}}\right )^{\prime }\left (p_n\right )} \) as a function of \(n \) are bounded below by the constant \(\frac {\propto l^2}{2ar}\), it follows that the values of the function \(\psi _n \) can be estimated from above as follows:

where \(h=\coth \left (\frac {a\tau ^2_0}{cl}\right )\). Taking into account the fact that \(\tau _n\ge \tau n\) (see Sec. 3), we conclude that \(\psi _n \) tends to zero as \(n\to \infty \). Consequently, the series \({R_\psi \left (t\right )} {=s\left (t\right )\sum ^{\infty }_{n=0}{\psi _n}\exp \left (p_nt\right ) } \) converges in the space \(\mathbf {L}_\mathbf {1}\left (\mathbf {R}^+\right )\) of functions of time integrable on \( \mathbf {R}^+=\left \{t\in \mathrm {Re}\thinspace :t>0\right \} \). The sum of this series is the impulse transfer function of the operator \(B_{\psi }\) with the transfer function \( \sum ^{\infty }_{n=0}{\frac {{\psi }_n}{p-p_n}} \).

Calculation of the expressions for \(b_0 \) , \( {b}_{j\mathrm{oc}}\) , \({ b}_{jn}\) \((j\!=\!1,2\) ; \(n\!\ge \!0) \) , \({ d}_k\) , \({ f}_k\) \( (k\!\ge \!1)\) , and \( \left (D_{\mathrm{T}}\right )^{\prime }\left (p_n\right ) \left (p_n\right ) \) .

1. 1.
   $$ b_0=\mathop {\mathrm {res}}\limits _{0}{ \Psi }_1=-\frac {1-\lim _{p\to 0} A_c(p)}{l\lim _{p\to 0} D_{\mathrm {T}}(p) }=\frac {r}{\propto l\left (r+2\right )}.$$

   (A.3.1)
2. 2.

   In view of the fact that \(p_{\mathrm{oc}}\) is a removable singularity of the function, we have

   $$ b_{2\mathrm{oc}}=\mathop {\mathrm {res}}\limits _{p_{\mathrm{oc}}}{ \Psi }_2=-\mathop {\mathrm {res}}\limits _{p_{\mathrm{oc}}}{ \Psi }_1=-b_{1\mathrm{oc}}.$$

   (A.3.2)
3. 3.
   $$ b_{1n}=\mathop {\mathrm {res}}\limits _{p_n}{ \Psi }_1=\frac {cl^2}{{\left (a\tau _n\right )}^2+{\left (cl\right )}^2} \frac {g_{1n}}{\left (D_{\mathrm{T}}\right )^{\prime }\left (p_n\right )}, $$

   (A.3.3)

   where

   $$ g_{1n}=\frac {{\cosh \left (a\tau ^2_nx/cl^2\right )-{\cosh \big (a\tau ^2_n\left (l-x\right )/cl^2\big )\big ({\cos \tau _n }+\left (r/\tau _n\right ){\sin \tau _n }\big ) } }}{{\sinh \left (a\tau ^2_n/cl\right ) }};$$
   $$ b_{2n}=\mathop {\mathrm {res}}\limits _{p_n}{ \Psi }_2=\frac {al}{{\left (a\tau _n\right )}^2+{\left (cl\right )}^2} \frac {g_{2n}}{\left (D_{\mathrm{T}}\right )^{\prime }\left (p_n\right )}, $$

   (A.3.4)

   where

   $$ g_{2n}=\tau _n{\sin \left (\frac {l-x}{l}\tau _n\right )-r{\cos \left (\frac {l-x}{l}\tau _n\right ) } }. $$
4. 4.

   To calculate the quantity \(\left (D_{\mathrm{T}}\right )^{\prime }\left (p_n\right )\) occurring in the expression for \(b_{jn} \) (\(j=1, 2 \)), we write (see the explanations to (3.2))

   $$ \begin {gathered} \frac {\partial D_{\mathrm {T}}}{\delta \zeta }=\alpha \left (2+\frac {1}{r}-\frac {r}{\zeta ^2}\right ){\sinh \zeta +\alpha }\left (\frac {\zeta }{r}+\frac {r}{\zeta }\right ){\cosh \zeta }, \\[.3em] \zeta ^{\prime }\left (p_n\right )=\frac {l}{2\sqrt {ap_n}}=-i\frac {l^2}{2a\tau _n}, \\[.3em] \left (D_{\mathrm {T}}\right )^{\prime }\left (p_n\right )=\frac {\partial D_{\mathrm {T}}}{\delta \zeta }\big (\zeta \left (p_n\right )\!\big )\zeta ^{\prime } \left (p_n\right ) {}=\frac {\propto l^2}{a\tau _n}\left [\left (1+\frac {1}{2r}+\frac {r}{2\tau ^2_n}\right ) {\sin \tau _n+\frac {1}{2} } \left (\frac {\tau _n}{r}-\frac {r}{\tau _n}\right ){\cos \tau _n }\right ]. \end {gathered}$$

   Taking into account relation (3.4) (see Sec. 3), which implies that

   $$ \sin \tau _n=2\frac {r\tau _n}{\tau ^2_n+r^2}\quad \text {and}\quad \cos \tau _n=\frac {\tau ^2_n-r^2}{\tau ^2_n+r^2},$$

   we finally obtain

   $$ \left (D_{\mathrm{T}}\right )^{\prime }\left (p_n\right )= \frac {\propto l^2}{a\left (\tau ^2_n+r^2\right )} \left [1+2r+\left (\frac {r}{\tau _n}\right )^2 +\frac {{\left (\tau ^2_n-r^2\right )}^2}{2r\tau ^2_n}\right ].$$

   (A.3.5)

   As can readily be verified, the values of the function \(\left (D_{\mathrm {T}}\right )^{\prime }\left (p_n\right )\) monotone decrease as \(n \) grows and hence exceed its value as \(n=\infty \), i.e., the quantity \(\frac {\propto l^2}{2ar} \).

5. 5.

   We start the calculation of the expressions for the coefficients \(d_k \) and \( f_k \) by calculating \(\mathop {\mathrm {res}}\limits _{i\omega _k}{\thinspace \Psi }_1\).

Since one has the expression

for \(\sigma \) (see the explanations to (5.3)), we obtain where

Thus, we obtain expressions for \(d_k\) and \(f_k \) (see the explanations to (A.2.12)) in the form

where

Here we have used the relation

1. Estimating the norm of the operator \( d^{-1} \): \(\left (\mathbf {QL}_1\right )_{\mathrm{y}}\to \mathbf {Q}_{\mathrm{y}} \).

If \( {f\in \left (\mathbf {QL}_1\right )}_{\mathrm{y}} \), then the function \(d^{-1}f \) is bounded, i.e., belongs to the space \(\mathbf {Q}_{\mathrm{y}}\), and its norm in this space does not exceed \( \int \nolimits ^{\infty }_0{\left |f\left (t\right )\right |}dt \). Therefore, the norm of \(d^{-1} \) as an operator \({\left (\mathbf {QL}_1\right )}_{\mathrm{y}}\to Q_{\mathrm{y}}\) does not exceed \( \vartheta _1\).

2. Estimating the norm of the operator \(B_{\Psi } \).

Majorizing the function \(\rho \left (\tau _n\right )=\sqrt {\tau ^2_n+r^2} \) in (A.2.16) for \(\tau _n\ge \tau _{0 } \) by the linear function of \(\tau _n \) of the form \(v\left (\tau _n\right )=\tau _n-\tau _0+\rho \left (\tau _0\right )= \tau _n+\frac {r^2}{\tau _0+\rho \left (\tau _0\right )}\), we can estimate the sum of the series \(R_\psi \left (t\right )\) from above (see Appendix 2) in the space \(L_{1\mathrm{y}} \) by the sum of the numerical series

where \({ s}_1=2h\frac {cl}{a}+\frac {r^2}{\tau _0+\rho \left (\tau _0\right )}\) and \({ s}_2=h\frac {cl}{a}\frac {r^2}{\tau _0+\rho \left (\tau _0\right )} \), \(h \); see the explanations to (A.2.15).

Using the formula \(\sum ^{\infty }_{n=1}{\frac {1}{n^4}=\frac {\pi ^4}{90}} \) (see [15, item 1.1.3.1, formula (16)]) and the inequality

we obtain the upper bound for \(M_\psi \), where \(h_1=2\frac {chl}{a}+\frac {r^2}{\tau _0+\rho }\), \(h_2=\frac {chl}{a}\frac {r^2}{\tau _0+\rho }\), \(h={\coth \left (\frac {a}{cl}\tau ^2_0\right )}\), \(\rho =\sqrt {r^2+\tau ^2_0}\), and \(\tau _0 \) is defined in Sec. 4.

The quantity \(M_B\) serves as an upper bound for the norm of the operator \({ B}_{\Psi }\).

3. Estimating the norm of the operator \({\Omega }_{\Psi }\): \(\mathbf {L}_{1\mathrm{y}}\to \mathbf {Q}_{\mathrm{y}}\).

The output of the operator \({\Omega }_{\Psi }\) with the input action \(u\) is determined by the convolution (see [16, Ch. 1, Sec. 4, item 7]) of the function \(u \) with the impulse transfer function \(w\left ({\Omega }_{\Psi }\right )\) of the operator \({\Omega }_{\Psi }\),

It follows from (A.4.3) that if the function \(w\left ({\Omega }_{\Psi }\right )\) belongs to the space \(\mathbf {Q}_{\mathrm{y}}\) and the function \(u \) belongs to the space \(\mathbf {L}_{1\mathrm{y}} \), then the output of the operator \({ \Omega }_{\Psi } \) belongs to the space \(\mathbf {Q}_{\mathrm{y}} \), its norm in this space not exceeding \({\left \|w\left ({\Omega }_{\Psi }\right )\right \|}_{\mathbf {Q}_{\mathrm{y}}}{\left \|u\right \|}_{\mathbf {L}_{1\mathrm{y}}} \).

Let us show that the function \(w\left ({\Omega }_{\Psi }\right ) \) belongs to the space \(\mathbf {L}_{1\mathrm{y}} \) and estimate its norm in this space based on the expression \( {w(\Omega _{\Psi })(t)}{=s(t)\sum ^{\infty }_{k=1}(d_k \cos \omega _k t+f_k \sin \omega _kt )}\) (see the statement of Theorem 3).

Each term on the right-hand side in this expression can be represented as \(M_k\thinspace {\sin \left (\omega _kt+\chi _k\right ) }\), where \(M_k=\sqrt {d^2_k+\omega ^2_k}\), \(\sin \chi _k=\frac {d_k}{M_k} \), and \(\cos \chi _k =\frac {f_k}{M_k} \). Consequently, this term belongs to the space \({\mathbf {Q}}_{\mathrm{y}}\) and has the norm equal to \(M_k \) in this space.

It follows from the representations (A.3.7) and (A.3.8) for \(d_k\) and \(f_{k} \), respectively, that

where

The quantity \(v_k \) is estimated from above by the quantity

To estimate the quantity \(M_k\) from above, we write out the expression for \({\left |D_{\mathrm {T}}\left (\zeta _k\right )\right |}^2 \) using the explanations to (A.3.6),

Since \(\sinh 2y_{k }>{2y_{k }> \sin {2y}_{k } } \), it follows that the right-hand side of (A.4.5) is positive and can be estimated from below by the function

Keeping only the term with the greatest growth rate in \(k \) in (A.4.6), we obtain a lower bound for the function \(\left |D_{\mathrm {T}}\left (\zeta _k\right )\right | \),

At the same time, the function \({\left |A_c\left (\zeta _k\right )\right |}^2 \) can be represented in the form

and estimated from above by the function while the function \(\left |\mathrm {Re}\thinspace A_c\left (\zeta _k\right )\right |=\left |h_{c{}ck}+\frac {1}{2}\frac {r}{y_k}\left (h_{csk}+h_{sck}\right )\right | \) is estimated from above by the function \(\left (1+\frac {1}{\sqrt {2}}\frac {r}{y_k}\right ){\cosh y_{k}} \).

Thus, the quantity \(M_k\) is estimated from above as follows:

where (\(\mu _k \) is a decreasing function of \(k \) that tends to 1 as \(k\to \infty \)).

The norm of the function \(w\left ({\Omega }_{\Psi }\right ) \) in the space \(\mathbf {Q}_{\mathrm{y}} \) is estimated by the sum of the numerical series \(\sum ^{\infty }_{k=1}{M_k}\).

Considering the estimate for the sum of the numerical series \(\sum ^{\infty }_{k=1}{\frac {1}{k^{3/2}}}\) from above by the quantity \(1+\int \nolimits ^{\infty }_0{\frac {dx}{x^{3/2}}}\) equal to 3, from (A.4.8) we conclude that \({\left \|w\left ({\Omega }_{\Psi }\right )\right \|_{\mathbf {Q}_{\mathrm{y}}}} \) is estimated from above by the quantity \(M_{\Omega }=6\frac {r\mu _1}{\alpha \pi ^{3/2}}\sqrt {\frac {c}{al}} \). If the function \(u \) belongs to the space \(\mathbf {L}_{1\mathrm{y}} \), then, by virtue of (A.4.3), we have

Consequently, the norm of the operator \({\Omega }_{\Psi } \): \({\left (\mathbf {QL}_1\right )}_{\mathrm{y}}\to \mathbf {Q}_{\Psi }\) does not exceed the ratio \(M_{\boldsymbol {\Omega }}\frac {{\left \|u\right \|}_{\mathbf {L}_{1\mathrm{y}}}}{\left \|u\right \|_{\left (\mathbf {QL}_1\right )_{\mathrm{y}}}}\), i.e., the quantity \( \vartheta _1M_{\boldsymbol {\Omega }}\) (see Sec. 5).

Thus, the norm of \(V_{\Psi }\) as of an operator \({\left ({\mathbf {QL}}_1\right )}_{\mathrm{y}}\to \mathbf {Q}_{\Psi } \) can be estimated from above by the quantity \(M_B+\vartheta _1\left (b_0+M_{\Omega }\right )\).

Proof of Theorem 4. The representation (6.4) for the operator \(V_\mathrm {M} \) follows from (5.3) and (5.4). By virtue of the commutation of the convolution operators [16, Ch. 1, Sec. 4, item 7] \(d^{-1} \) and \(V_{\Psi } \), we have

Therefore, if \(u\in \mathbf {U} \) (see Sec. 6), then its primitive \(d^{-1}u \) belongs to \({\left (\mathbf {QL}_1\right )}_{\mathrm{y}}\) and the function \(d^{-1}\left (V_{\Psi }\left (u\right )\right )\) belongs to the space \( \mathbf {Q}_{\Psi }\) (see the end of Sec. 5).

Thus, by (6.4), the operator \(V_{\mathrm{m}}\) maps the space \(\mathbf {U} \) into \(\mathbf {Q}_{\mathrm{m}} \) (see Sec. 6), and its norm, calculated as

is estimated from above by the quantity \(\beta \vartheta _2\left [M_B+\vartheta _1 \left (b_0+M_{\Omega }\right )\right ]\).

Proof of Theorem 5 .

Under conditions (7.3) and (7.5), by virtue of the operator \(V \) being linear, we obtain inequality (7.2) with constant \(L_X=L_F\left \|V\right \| \) strictly less than 1 and independent of \(y\in \mathbf {Q}\) and \(f\in F \). Therefore, the mapping \(X_f=V\circ F \) is a contraction and hence has a fixed point; it is this fact that implies the existence and uniqueness of a solution of system (2.1)–(2.3), (7.1) for each \(f\in F \), i.e., the existence of an operator \(A:\mathbf {F}\to \mathbf {Q}_2 \) that sends each exogenous action \(f \) to the solution of this system for \(y \). The upper bound for the norm of this operator can be obtained from (7.3) and (7.4), because for \(y=A\left (f\right ) \) one has the relations

and since \(L_X<1 \), we obtain