Proof of Theorem 1. Let \(w_{1}^{ki} \), \( \mathrm {v}_{1}^{i} \) and \(w_{2}^{ki} \), \(\mathrm {v}_{2}^{i} \) be arbitrary admissible parameters in
\(W^{ki} \times V^{i} \), and let \(u_{1}^{ki} (x) \) and \(u_{2}^{ki} (x) \) be the corresponding solutions of the boundary
value problems (2.1), (2.2); that is,
$$ \begin {gathered} \frac {du_{1}^{ki} (x)}{dx} =A^{ki}
(x)u_{1}^{ki} (x)+B^{ki} w_{1}^{ki} \delta (x-\xi ^{ki} )+f^{ki} (x),\\ x\in
(0,l^{ki} ),\quad k\in I_{i}^{+} ,\quad i\in I, \end {gathered}
$$
(A.1)
$$ \sum _{s=1,\thinspace k_{s} \in I_{i}^{-} }^{\underline
{n}_{i} }g_{j}^{ik_{s} } u_{1}^{ik_{s} } (0)+\sum _{s=1,\thinspace k_{s} \in
I_{i}^{+} }^{\overline {n}_{i} }q_{j}^{k_{s} i} u_{1}^{k_{s} i} (l^{k_{s} i} )
=\mathrm {v}_{1j}^{i} ,\quad j={1,\ldots ,M_{i}},\quad i\in I,
$$
(A.2)
$$ \begin {gathered} \frac {du_{2}^{ki} (x)}{dx} =A^{ki}
(x)u_{2}^{ki} (x)+B^{ki} w_{2}^{ki} \delta (x-\xi ^{ki} )+f^{ki} (x),\\ x\in
(0,l^{ki} ),\quad k\in I_{i}^{+} ,\quad i\in I, \end {gathered}
$$
(A.3)
$$ \begin {gathered} \sum _{s=1,\thinspace k_{s} \in
I_{i}^{-} }^{\underline {n}_{i} }g_{j}^{ik_{s} } u_{2}^{ik_{s} } (0)+\sum
_{s=1,\thinspace k_{s} \in I_{i}^{+} }^{\overline {n}_{i} }q_{j}^{k_{s} i}
u_{2}^{k_{s} i} (l^{k_{s} i} ) =\mathrm {v}_{2j}^{i} ,\\ j={1,\ldots ,M_{i}},\quad
i\in I.\end {gathered}$$
(A.4)
By virtue of the convexity of
the admissible set
\(\Omega _{{\rm v}^{i} } \), we have
\({w_{1}^{ki} =\lambda w_{1}^{ki} +(1-\lambda )w_{1}^{ki} \in \Omega _{w^{ki} }} \) for an arbitrary
\(\lambda \in [0;1] \). Set
\(u^{ki} (t)=\lambda u_{1}^{ki} (x)+(1-\lambda )u_{2}^{ki} (x)\).
Let us multiply both sides of (A.1) by \(\lambda \) and of
(A.3) by \((1-\lambda ) \), add the resulting equalities term by term, and
arrange the terms,
$$
\begin {aligned} \lambda \frac {du_{1}^{ki} (x)}{dx} +(1-\lambda )\frac
{du_{2}^{ki} (x)}{dx} &=A^{ki} (x)\left [\lambda u_{1}^{ki} (x)+(1-\lambda
)u_{2}^{ki} (x)\right ] \\ &\qquad \qquad {}+B^{ki} \left [\lambda w_{1}^{ki} +
(1-\lambda )w_{2}^{ki} \right ]\delta (x-\xi ^{ki} )\\ &\qquad \qquad \qquad
\qquad {}+\left [\lambda f^{ki} (x)+(1-\lambda )f^{ki} (x)\right ],\quad x\in
(0,l^{ki} ). \end {aligned}$$
It follows that the pair
\(\left (u^{ki} (x),w^{ki} \right )\) satisfies the system of
differential equations
$$
\begin {gathered} \frac {du^{ki} (x)}{dx} =A^{ki} (x)u^{ki} (x)+B^{ki} w^{ki}
\delta (x-\xi ^{ki} )+f^{ki} (x),\\ x\in (0,l^{ki} ), \quad k\in I_{i}^{+} ,\quad
i\in I. \end {gathered}$$
Multiplying both sides of
(
A.2) by
\(\lambda \) and of (
A.4) by
\((1-\lambda )\) and
performing summation and grouping, we obtain
$$ \begin {gathered} \sum _{s=1,\thinspace k_{s} \in
I_{i}^{-} }^{\underline {n}_{i} }g_{j}^{ik_{s} } \left [\lambda u_{1}^{ik_{s} }
(0)+(1-\lambda )u_{2}^{ik_{s} } (0)\right ]+\sum _{s=1,\thinspace k_{s} \in
I_{i}^{+} }^{\overline {n}_{i} }q_{j}^{k_{s} i} \left [\lambda u_{1}^{k_{s} i}
(l^{k_{s} i} )+(1-\lambda )u_{2}^{k_{s} i} (l^{k_{s} i} )\right ] \\ {}=\lambda
\mathrm {v}_{1j}^{i} +(1-\lambda )\mathrm {v}_{2j}^{i} , \quad j={1,\ldots
,M_{i}}, \quad i\in I. \end {gathered}$$
It follows
that the pair
\( \left (u^{ki} (x),\mathrm {v}^{i} \right )\)
satisfies the conditions
$$
\sum _{s=1,k_{s} \in I_{i}^{-} }^{\underline {n}_{i} }g_{j}^{ik_{s} } u^{ik_{s} }
(0)+\sum _{s=1,k_{s} \in I_{i}^{+} }^{\overline {n}_{i} }q_{j}^{k_{s} i} u^{k_{s}
i} (l^{k_{s} i} ) =\mathrm {v}_{j}^{i} , \quad j={1,\ldots ,M_{i}}, \quad i\in I.
$$
By virtue of the convexity of the functions \(f_{0}^{ki} (u^{ki} ,x) \) and \(\Phi (\underline {u},\overline {u},w,\mathrm {v},\xi )\) with respect to the
arguments \(u \), \(w \), and \(\mathrm {v} \), we have
$$ \begin {aligned} \mathfrak {I}\thinspace (w,{\rm v}, \xi
)&=\mathfrak {I}\thinspace \big (\lambda w_{1} +(1-\lambda )w_{2} ,\lambda {\rm
v}_{1} +(1-\lambda ){\rm v}_{2} ,\xi \big ) \\ &{}=\sum _{i\in I}\sum _{k\in
I_{i}^{+} }\int \limits _{0}^{l^{ki} }f_{0}^{ki} \big (\lambda u^{ki}
(x)+(1-\lambda )u^{ki} (x),x\big )dx \\ &\qquad {}+\Phi \big (\lambda \underline
{u}+(1-\lambda )\underline {u},\lambda \overline {u}+(1-\lambda )\overline
{u},\lambda w+(1-\lambda )w,\lambda {\rm v}+(1-\lambda ){\rm v}, \xi \big ) \\
&{}\le \sum _{i\in I}\sum _{k\in I_{i}^{+} }\left (\lambda \int \limits
_{0}^{l^{ki} }f_{0}^{ki} \big (u^{ki} (x),x\big )dx+ \right . (1-\lambda )\left .
\int \limits _{0}^{l^{ki} }f_{0}^{ki} \big (u^{ki} (x),x\big )dx \right ) \\
&\qquad {}+\lambda \Phi (\underline {u},\overline {u},w,{\rm v}, \xi )+(1-\lambda
)\Phi (\underline {u},\overline {u},w,{\rm v}, \xi ) \\[.3em] &{}\le \lambda
\mathfrak {I}\thinspace (w,{\rm v}, \xi )+(1-\lambda )\mathfrak {I}\thinspace
(w,{\rm v}, \xi ). \end {aligned}$$
(A.5)
This implies the convexity of the functional \(\mathfrak {I}\thinspace (w,{\rm v}, \xi )\) with respect to
\(w \) and \(\rm v \). It is clear that if one of the functions
\(f_{0}^{ki} (u^{ki} ,x)\) and \(\Phi (\underline {u},\overline {u},w,\mathrm {v},\xi )\) is strictly convex, then
the inequality in (A.5) will be strict.
Consequently, so will be the functional of problem (2.1), (2.2),
(2.4), (2.5) with respect to \(w \) and \(\rm v \). The proof of Theorem 1 is complete. \(\quad \blacksquare \)
Proof of Theorem 2. Using the increment
method for the vector to be optimized, let us prove the differentiability of the functional and
determine the linear parts of its increment [10, 11].
In system (2.1), we introduce the
notation \(W^{ki} =W^{ki} (x;w^{ki} ,\xi ^{ki})=w^{ki} \delta (x-\xi ^{ki} ) \), \( (k,i)\in J \), and \(W=W(x;w,\xi )=\left (W^{ki} (x,w^{ki} ,\xi ^{ki} ):\thinspace (k,i)\in J\right ) \) for the terms to be optimized and write
system (2.1) as
$$ \frac {du^{ki} (x)}{dx}
=A^{ki} (x)u^{ki} (x)+B^{ki} W^{ki} (x;w^{ki} ,\xi ^{ki} )+f^{ki} (x), \quad (k,
i)\in J.$$
Assume that the parameter triple \(\upsilon =(w,{\rm v}, \xi ) \) to be optimized has received an increment
\(\Delta \upsilon =(\Delta w,\Delta {\rm v}, \Delta \xi ) \). Set \(\tilde {\upsilon }=\upsilon +\Delta \upsilon \), \( \tilde {w}=w+\Delta w \), \( \tilde {{\rm v}}={\rm v}+\Delta {\rm v}\), and \({ \tilde {\xi }=\xi +\Delta \xi } \).
In this case, the function \(W(x;w,\xi )\) and the
solution of the boundary value problem (2.1),
(2.2) will receive the increments
$$ \begin {gathered} \Delta
W(x;w,\xi )=W(x;w+\Delta w,\xi +\Delta \xi )-W(x;w,\xi ), \\ \Delta u^{ki}
(x;\upsilon )=u^{ki} (x;\upsilon +\Delta \upsilon )-u^{ki} (x;\upsilon )=\tilde
{u}^{ki} (x;\tilde {\upsilon })-u^{ki} (x;\upsilon ), \quad (k,i)\in J, \end
{gathered}$$
where the
\(u^{ki} (x;\upsilon ) \) and
\(\tilde {u}^{ki} (x;\tilde {\upsilon })\),
\((k,i)\in J \), are the solutions of the boundary value
problems (
2.1) and (
2.2) with the parameters
\(\upsilon \) and
\(\tilde {\upsilon }=\upsilon +\Delta \upsilon \), respectively, to be optimized.
It can readily be shown that the \(\Delta u^{ki} (x;\upsilon ) \), \( (k,i)\in J \), are solutions of the system of boundary value
problems
$$ \frac {d\Delta u^{ki} (x)}{dx} =A^{ki} (x)\Delta u^{ki}
(x)+B^{ki} \Delta W^{ki} (x;w^{ki} ,\xi ^{ki} ), \quad (k, i)\in J,
$$
(A.6)
$$ \sum _{s=1,\thinspace k_{s} \in I_{i}^{-} }^{\underline
{n}_{i} }g_{j}^{ik_{s} } \Delta u^{ik_{s} } (0)+\sum _{s=1,\thinspace k_{s} \in
I_{i}^{+} }^{\overline {n}_{i} }q_{j}^{k_{s} i} \Delta u^{k_{s} i} (l^{k_{s} i} )
=\Delta \mathrm {v}_{j}^{i} ,\\ j={1,\ldots ,M_{i}},\quad i\in I.
$$
(A.7)
Then for the increment of the functional (2.5) we have
$$ \begin {aligned} \Delta \mathfrak {I}\thinspace
(w,\mathrm {v},\xi )&=\mathfrak {I}\thinspace (\hat {w},\hat {\mathrm {v}},\hat
{\xi })-\mathfrak {I}\thinspace (w,\mathrm {v},\xi )=\sum _{i\in I}\sum _{k\in
I_{i}^{+} }\int \limits _{0}^{l^{ki} }\frac {\partial f_{0}^{ki} }{\partial u^{ki}
} \Delta u^{ki} (x) \\ &\qquad {}+\frac {\partial \Phi }{\partial \underline {u}}
\Delta \underline {u}+\frac {\partial \Phi }{\partial \overline {u}} \Delta
\overline {u}+\frac {\partial \Phi }{\partial w} \Delta w+\frac {\partial \Phi
}{\partial {\rm v}} \Delta {\rm v+}\frac {\partial \Phi }{\partial \xi } \Delta
\xi +\eta , \end {aligned}$$
(A.8)
$$ \eta =o\left (\big \| \Delta u(x)\big \| _{L_{2}^{M}
[0,l]} ,\left \| \Delta \underline {u}\right \| _{{\rm R}^{\underline {n}} }
,\left \| \Delta \overline {u}\right \| _{{\rm R}^{\overline {n}} } ,\left \|
\Delta w\right \| _{{\rm R}^{\mu } } ,\left \| \Delta \xi \right \| _{{\rm
R}^{\overline {m}} } ,\left \| \Delta {\rm v}\right \| _{{\rm R}^{M} } \right ).
$$
(A.9)
Here we have denoted
\(f_{0}^{ki} =f_{0}^{ki} (u^{ki}(x),x)\) and
\(\Phi =\Phi (\underline {u},\overline {u},w,\mathrm {v},\xi )\), and
\(\eta \) is the residual term in appropriate spaces of
functions and finite-dimensional vectors.
It is well known [24] in the
theory of differential equations that under the assumptions made about the data occurring in the
problem one has the estimate
$$ \big \| \Delta u(x)\big \|_{L_{2}^{M} [0,l]} \le O\left
(\left \| \Delta w\right \| _{{\rm R}^{\mu } } ,\left \| \Delta \xi \right \|
_{{\rm R}^{\overline {m}} } ,\left \| \Delta {\rm v}\right \| _{{\rm R}^{M} }
\right ),$$
and hence the following estimates hold:
$$ \begin {gathered}
{\left \| \Delta \underline {u}\right \| _{{\rm R}^{\underline {n}} } =\big \|
\Delta u(0)\big \| _{{\rm R}^{\underline {n}} } \le O\big (\left \| \Delta w\right
\| _{{\rm R}^{\mu } } ,\left \| \Delta \xi \right \| _{{\rm R}^{\overline {m}} }
,\left \| \Delta {\rm v}\right \| _{{\rm R}^{M} } \big ),} \\ {\left \| \Delta
\overline {u}\right \| _{{\rm R}^{\overline {n}} } =\big \| \Delta u(l)\big \|
_{{\rm R}^{\overline {n}} } \le O\big (\left \| \Delta w\right \| _{{\rm R}^{\mu }
} ,\left \| \Delta \xi \right \| _{{\rm R}^{\overline {m}} } ,\left \| \Delta {\rm
v}\right \| _{{\rm R}^{M} } \big ).} \end {gathered}
$$
Then from (A.9) we have the
main estimate
$$ \begin
{gathered} \eta =o\big (\left \| \Delta w\right \| _{{\rm R}^{\mu } } ,\left \|
\Delta \xi \right \| _{{\rm R}^{\overline {m}} } ,\left \| \Delta {\rm v}\right \|
_{{\rm R}^{M}}\big ), \end {gathered}$$
which implies
the differentiability of the functional
\(\mathfrak {I}\thinspace (w,{\rm v}, \xi ) \) with respect to all of its arguments.
Now let us present formulas for the components of the gradient of the functional of
the problem with respect to \(w\), \(\mathrm {v} \), and \(\xi \). To this end, we transfer the right-hand sides of
Eqs. (A.6) to the left and multiply the relations
by as yet arbitrary \(\aleph \)-dimensional
vector functions \( \psi ^{ki} (x)\in \mathrm {R}^{\aleph } \),
\(x\in (0,l^{ki} )\), \(k\in I_{i}^{+} \), \( i\in I \), continuously differentiable with respect to their
arguments. Let us add up the resulting expressions, equate the sum with zero, and integrate by
parts in the sum,
$$ \begin {aligned} 0 &= \sum _{i\in I}\sum _{k\in I_{i}^{+}
}\int \limits _{0}^{l^{ki} }\left [\big (\psi ^{ki} (x)\big )^{\mathrm {T}} \left
( \frac {d\Delta u^{ki} (x)}{dx} - A^{ki} (x)\Delta u^{ki} (x) - B^{ki} \Delta
W^{ki} (x,\Delta w,\Delta \xi ) \right ) \right ]dx \\ &{}=\sum _{i\in I}\left
\{\left [\sum _{k\in I_{i}^{+} }\big (\psi ^{ki} (l^{ki} )\big )^{\mathrm {T}}
\Delta u^{ki} (l^{ki} ) \right . \right . \left . -\sum _{k\in I_{i}^{-} }\big
(\psi ^{ik} (0)\big )^{\mathrm {T}} \Delta u^{ik} (0) \right ] \\ &\qquad \qquad
\qquad \qquad \qquad {}-\left . \sum _{k\in I_{i}^{+} }\int \limits _{0}^{l^{ki}
}\left [\left (\left (\frac {d\psi ^{ki} (x)}{dx} \right )^{\mathrm {T}} +\big
(\psi ^{ki} (x)\big )^{\mathrm {T}} A^{ki} (x)\right )\Delta u^{ki} (x)\right
.\right . \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad {} \left .\left .\vphantom {\left [\left (\left (\frac {d\psi ^{ki}
(x)}{dx} \right )^{\mathrm {T}} +(\psi ^{ki} (x))^{\mathrm{T}} A^{ki} (x)\right
)\Delta u^{ki} (x)-\right .}{}-\big (\psi ^{ki} (x)\big )^{\mathrm {T}} B^{ki}
\Delta W^{ki} (x,\Delta w,\Delta \xi )\right ] dx\right \}. \end {aligned}
$$
(A.10)
Let us add the right-hand side of (
A.10) to (
A.8)
and arrange terms to obtain
$$ \begin {aligned} &{}\Delta \mathfrak {I}\thinspace
(w,{\rm v}, \xi ) =\sum _{i\in I}\sum _{k\in I_{i}^{+} }\int \limits _{0}^{l^{ki}
}\left \{\left [\frac {\partial f_{0}^{ki} }{\partial u^{ki} } -\left (\frac
{d\psi ^{ki} (x)}{dx} \right )^{\mathrm {T}} -\big (\psi ^{ki} (x)\big )^{\mathrm
{T}} A^{ki} (x)\right ] \Delta u^{ki} (x) \right . \\[.3em] &\qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad {}\left .\vphantom {\left \{\left [\frac
{\partial f_{0}^{ki} }{\partial u^{ki} } -\left (\frac {d\psi ^{ki} (x)}{dx}
\right )^{\mathrm {T}} -(\psi ^{ki} (x))^{\mathrm {T}} A^{ki} (x)\right ] \Delta
u^{ki} (x) -{}\right .} {} -\big (\psi ^{ki} (x)\big )^{\mathrm {T}} B^{ki} \Delta
W^{ki} (x,\Delta w,\Delta \xi )\right \}dx \\[.3em] &\qquad {}+ \left ( \frac
{\partial \Phi }{\partial \underline {u}} - (\underline {\psi })^{\mathrm {T}}
\right ) \Delta \underline {u} + \left ( \frac {\partial \Phi }{\partial \overline
{u}} + (\overline {\psi })^{\mathrm {T}} \right ) \Delta \overline {u} + \frac
{\partial \Phi }{\partial w} \Delta w + \frac {\partial \Phi }{\partial {\rm v}}
\Delta {\rm v} + \frac {\partial \Phi }{\partial \xi } \Delta \xi + \eta , \end
{aligned}$$
(A.11)
where
$$ \begin {gathered} \underline
{\psi }=\left (\underline {\psi }^{i} :i\in I\right )\in {\rm R}^{\underline
{n}},\quad \overline {\psi }=\left (\overline {\psi }^{i} :i\in I\right )\in {\rm
R}^{\overline {n}},\\[.3em] \begin {aligned} \underline {\psi }^{i} &=\big (\psi
^{ik_{1} } (0),\ldots ,\psi ^{ik_{\underline {n}_{i} } } (0)\big )^{\mathrm
{T}},\\ \overline {\psi }^{i} &=\big (\psi ^{k_{1} i} (l^{k_{1} i} ),\ldots ,\psi
^{k_{\overline {n}_{i} } i} (l^{k_{\overline {n}_{i} } i})\big )^{\mathrm {T}}.
\end {aligned} \end {gathered}$$
Let us transform the
second term in the integrand of (
A.11) neglecting
terms that are less than the first order of smallness,
$$ \begin {aligned} &\sum _{i\in I}\sum _{k\in I_{i}^{+}
}\int \limits _{0}^{l^{ki} }\big (\psi ^{ki} (x)\big )^{\mathrm {T}} B^{ki} \Delta
W^{ki} (x,\Delta w,\Delta \xi )dx \\[.4em] &\qquad {}= \sum _{i\in I}\sum _{k\in
I_{i}^{+} }\int \limits _{0}^{l^{ki} } \big (\psi ^{ki}(x)\big )^{\mathrm {T}}
B^{ki} \Big [ (w^{ki} + \Delta w^{ki} )\delta \big (x - (\xi ^{ki} + \Delta \xi
^{ki} )\big ) - w^{ki} \delta (x - \xi ^{ki} ) \Big ] dx \\[.4em] &\qquad {}=\sum
_{i\in I}\sum _{k\in I_{i}^{+} }\left (\int \limits _{0}^{l^{ki} }\big (\psi ^{ki}
(x)\big )^{\mathrm {T}} \delta \big (x-(\xi ^{ki} +\Delta \xi ^{ki} )\big )
dx\right ) B^{ki} (w^{ki} +\Delta w^{ki} ) \\ &\qquad \qquad {}-\sum _{i\in I}\sum
_{k\in I_{i}^{+} }\left (\int \limits _{0}^{l^{ki} }\big (\psi ^{ki} (x)\big
)^{\mathrm {T}} \delta (x-\xi ^{ki} )dx \right ) B^{ki} w^{ki} \\[.4em] &\qquad
{}=\sum _{i\in I}\sum _{k\in I_{i}^{+} }\big (\psi ^{ki} (\xi ^{ki} +\Delta \xi
^{ki} )\big )^{\mathrm {T}} B^{ki} (w^{ki} +\Delta w^{ki} ) -\sum _{i\in I}\sum
_{k\in I_{i}^{+} }\big (\psi ^{ki} (\xi ^{ki} )\big )^{\mathrm {T}} B^{ki} w^{ki}
\\[.4em] &\qquad {}=\sum _{i\in I}\sum _{k\in I_{i}^{+} }\big (\psi ^{ki} (\xi
^{ki} +\Delta \xi ^{ki} )\big )^{\mathrm {T}} B^{ki} w^{ki} +\sum _{i\in I}\sum
_{k\in I_{i}^{+} }\big (\psi ^{ki} (\xi ^{ki} +\Delta \xi ^{ki} )\big )^{\mathrm
{T}} B^{ki} \Delta w^{ki} \\[.4em] &\qquad \qquad {}-\sum _{i\in I}\sum _{k\in
I_{i}^{+} }\big (\psi ^{ki} (\xi ^{ki} )\big )^{\mathrm {T}} B^{ki} w^{ki}
\\[.4em] &\qquad {}=\sum _{i\in I}\sum _{k\in I_{i}^{+} }\left (\left . \frac
{d\psi ^{ki} (x)}{dx} \right |_{x=\xi ^{ki} } \right )^{\mathrm {T}} B^{ki} w^{ki}
\Delta \xi ^{ki} +\sum _{i\in I}\sum _{k\in I_{i}^{+} }\big (\psi ^{ki} (\xi ^{ki}
)\big )^{\mathrm {T}} B^{ki} \Delta w^{ki} . \end {aligned}
$$
Substituting the resulting expression into (
A.11), after rearranging the terms we obtain
$$ \begin {aligned} \Delta \mathfrak {I}\thinspace (w,{\rm
v}, \xi )&=\sum _{i\in I}\sum _{k\in I_{i}^{+} }\int \limits _{0}^{l^{ki} }\left
[\frac {\partial f_{0}^{ki} }{\partial u^{ki} } -\left (\frac {d\psi ^{ki}
(x)}{dx} \right )^{\mathrm {T}} -\big (\psi ^{ki} (x)\big )^{\mathrm {T}} A^{ki}
(x)\right ]\Delta u^{ki} (x)dx \\ &\quad {}-\sum _{i\in I}\sum _{k\in I_{i}^{+}
}\left [\left (\left . \frac {d\psi ^{ki} (x)}{dx} \right |_{x=\xi ^{ki} } \right
)^{\mathrm {T}} B^{ki} w^{ki} -\frac {\partial \Phi }{\partial \xi ^{ki} } \right
] \Delta \xi ^{ki} \\ &\quad {}-\sum _{i\in I}\sum _{k\in I_{i}^{+} }\left [\left
(\psi ^{ki} (\xi ^{ki} )\right )^{\mathrm {T}} B^{ki} -\frac {\partial \Phi
}{\partial w^{ki} } \right ] \Delta w^{ki} +\sum _{i\in I}\frac {\partial \Phi
}{\partial {\rm v}^{i} } \Delta {\rm v}^{i} \\ &\quad {}+\left [\left (\frac
{\partial \Phi }{\partial \underline {u}} -(\underline {\psi })^{\mathrm {T}}
\right )\Delta \underline {u}+\left (\frac {\partial \Phi }{\partial \overline
{u}} +(\overline {\psi })^{\mathrm {T}} \right )\Delta \overline {u}\right ]+\eta
. \end {aligned}$$
(A.12)
Let us deal with the expression in the last brackets.
To simplify the presentation of the calculations to follow, instead of matrix and
vector operations we also use their componentwise notation. We write conditions (A.7) in the form
$$ \begin {aligned} &\left (\begin {array}{rcl}
g_{1{}1}^{ik_{1} } \ldots g_{1\aleph }^{ik_{1} }& \ldots & g_{1\underline {n}_{i}
}^{ik_{\underline {n}_{i} } } \ldots g_{1\underline {n}_{i} \cdot \aleph
}^{ik_{\underline {n}_{i} } } \\ \ldots & \ldots & \ldots \\ g_{M_{i} ,1}^{ik_{1}
} \ldots g_{M_{i} ,\aleph }^{ik_{1} } & \ldots & g_{M_{i} ,\underline {n}_{i}
}^{ik_{\underline {n}_{i} } } \ldots g_{M_{i} \underline {,n}_{i} \cdot \aleph
}^{ik_{\underline {n}_{i} } } \end {array}\right )\left (\begin {array}{c} {\Delta
\underline {u}_{1}^{i} } \\ {\ldots } \\ {\Delta \underline {u}_{\underline
{n}_{i} \cdot \aleph }^{i} } \end {array}\right ) \\[.3em] &\qquad \qquad {}+\left
(\begin {array}{rcl} q_{1{}1}^{ik_{1} } \ldots q_{1\aleph }^{ik_{1} }& \ldots &
q_{1\overline {n}_{i} }^{ik_{\overline {n}_{i} } } \ldots q_{1\overline {n}_{i}
\cdot \aleph }^{ik_{\overline {n}_{i} } } \\ \ldots & \ldots & \ldots \\ q_{M_{i}
,1}^{ik_{1} } \ldots q_{M_{i} ,\aleph }^{ik_{1} }& \ldots & q_{M_{i} ,\overline
{n}_{i} }^{ik_{\overline {n}_{i} } } \ldots q_{M_{i} ,\overline {n}_{i} \cdot
\aleph }^{ik_{\overline {n}_{i} } } \end {array}\right )\left (\begin {array}{c}
{\Delta \overline {u}_{1}^{i} } \\ {\ldots } \\ {\Delta \overline {u}_{\overline
{n}_{i} \cdot \aleph }^{i} } \end {array}\right )=\left (\begin {array}{c} {\Delta
{\rm v}_{1}^{i} } \\ {\ldots } \\ {\Delta {\rm v}_{M_{i} }^{i} } \end
{array}\right ). \end {aligned}$$
In the above notation, relations (A.7) acquire the form
$$ \left (\begin {array}{{ccc}} {c_{1,1}^{i} } & {\ldots } &
{c_{1,(n_{i} \cdot \aleph )}^{i} } \\ {\ldots } & {\ldots } & {\ldots } \\
{c_{M_{i} ,1}^{i} } & {\ldots } & {c_{M_{i} ,(n_{i} \cdot \aleph )}^{i} } \end
{array} \right )\left (\begin {array}{c} {\Delta u_{1}^{i} } \\ {\ldots } \\
{\Delta u_{n_{i} \cdot \aleph }^{i} } \end {array}\right )=\left (\begin
{array}{c} {\Delta {\rm v}_{1}^{i} } \\ {\ldots } \\ {\Delta {\rm v}_{M_{i} }^{i}
} \end {array}\right ),\quad i\in I,$$
(A.13)
or, in matrix
form,
$$ C_{i} \Delta
u^{i} =\Delta \mathrm {v}^{i} ,\quad i\in I.$$
Then (A.13) can be written as
$$ \begin {aligned}
&\left (\begin {array}{{ccc}} {\stackrel {_\frown }{c}{\!}_{1,1}^{i} } & {\ldots }
& {\stackrel {_\frown }{c}{\!}_{1,M_{i} }^{i} } \\ {\ldots } & {\ldots } & {\ldots
} \\ {\stackrel {_\frown }{c}{\!}_{M_{i} ,1}^{i} } & {\ldots } & {\stackrel
{_\frown }{c}{\!}_{M_{i} ,M_{i} }^{i} } \end {array} \right )\left (\begin
{array}{c} {\Delta \stackrel {_\frown }{u}{\!}_{1}^{i} } \\ {\ldots } \\ {\Delta
\!\stackrel {_\frown }{u}{\!}_{M_{i} }^{i} } \end {array}\right ) \\[.3em] &\qquad
\qquad {}+\left (\begin {array}{{ccc}} {\stackrel {_\smile }{c}{\!}_{1,1}^{i} } &
{\ldots } & {\stackrel {_\smile }{c}{\!}_{1,(n_{i} \cdot \aleph -M_{i} )}^{i} } \\
{\ldots } & {\ldots } & {\ldots } \\ {\stackrel {_\smile }{c}{\!}_{M_{i} ,1}^{i} }
& {\ldots } & {\stackrel {_\smile }{c}{\!}_{M_{i} ,(n_{i} \cdot \aleph -M_{i}
)}^{i} } \end {array} \right )\left (\begin {array}{c} {\Delta \stackrel {_\smile
}{u}{\!}_{1}^{i} } \\ {\ldots } \\ {\Delta \!\stackrel {_\smile }{u}{\!}_{(n_{i}
\cdot \aleph )-M_{i} }^{i} } \end {array}\right )=\left (\begin {array}{c} {\Delta
{\rm v}_{1}^{i} } \\ {\ldots } \\ {\Delta {\rm v}_{M_{i} }^{i} } \end
{array}\right ), \end {aligned}$$
or in the form
$$ \stackrel {_\frown }{C}_{i} \Delta \stackrel {_\frown
}{u}{\!}^{i} +\stackrel {_\smile }{C}_{i} \Delta \stackrel {_\smile }{u}{\!}^{i}
=\Delta \mathrm {v}^{i} ,\quad i\in I.$$
(A.14)
In view
of (
A.13),
\(\stackrel {_\frown }{C}_{i} \) has an inverse matrix. Then from (
A.14) we have
$$ \Delta \!\stackrel {_\frown }{u}{\!}^{i} =-\left
(\stackrel {_\frown }{C}_{i} \right )^{-1} \stackrel {_\smile }{C}_{i} \Delta
\stackrel {_\smile }{u}{\!}^{i} +\stackrel {_\frown }{C}{\!}_{i} ^{-1} \Delta
\mathrm {v}^{i} ,\quad i\in I.$$
(A.15)
According to (A.15), we take the
\(M_{i} \)-dimensional increment vector
$$ {\Delta \!\stackrel {_\frown
}{u}{\!}^{i}= (\Delta \stackrel {_\frown }{u}{\!}_{1}^{i} ,\ldots ,\Delta
\stackrel {_\frown }{u}{\!}_{M_{i} }^{i} )^{\mathrm {T}} =\left (\Delta
u_{\stackrel {_\frown }{\mu }_{1} }^{i} ,\ldots ,\Delta u^{i}_{\stackrel {_\frown
}{\mu }_{M_{i} } } \right )^{\mathrm {T}} }$$
to be
dependent, and the
\((n_{i} \aleph -M_{i} ) \)-dimensional vector
$$ {\Delta \!\stackrel {_\smile
}{u}{\!}^{i} =(\Delta \stackrel {_\smile }{u}{\!}_{1}^{i} ,\ldots , \Delta
\stackrel {_\smile }{u}{\!}_{n_{i} \cdot \aleph -M_{i} }^{i} )^{\mathrm {T}}
=\left (\Delta u_{\stackrel {_\smile }{\mu }_{1} }^{i} ,\ldots ,\Delta
u_{\stackrel {_\smile }{\mu } _{{n_{i}\cdot \aleph -M_{i} }}} ^{i} \right
)^{\mathrm {T}} }$$
to be independent. Let us take into
account (
A.15) in the expression in the last
brackets in (
A.12) to obtain
$$ \begin {aligned} &\sum
_{i\in I}\left (\left (\frac {\partial \Phi }{\partial \underline {u}^{i} }
-(\underline {\psi }^{i} )^{\mathrm{T}} \right )\Delta \underline {u}^{i} +
\left (\frac {\partial \Phi }{\partial \overline {u}^{i} } +(\overline {\psi
}{}^{i} )^{\mathrm{T}} \right )\Delta \overline {u}{}^{\thinspace i} \right )=
\sum _{i\in I}\left (\frac {\partial \Phi }{\partial u^{i} } +(\psi ^{i}
)^{\mathrm {T}} \right )\Delta u^{i}\\ &\qquad {}=\sum _{i\in I}\left (\frac
{\partial \Phi }{\partial \stackrel {_\frown }{u}{\!}^{i} } +(\stackrel {_\frown
}{\psi }{\!}^{i} )^{\mathrm {T}} \right )\Delta \stackrel {_\frown }{u}{\!}^{i}
+\sum _{i\in I}\left (\frac {\partial \Phi }{\partial \stackrel {_\smile
}{u}{\!}^{i} } +(\stackrel {_\smile }{\psi }{\!}^{i} )^{\mathrm {T}} \right
)\Delta \stackrel {_\smile }{u}{\!}^{i} \\ &\qquad {}=\sum _{i\in I}\left (\frac
{\partial \Phi }{\partial \stackrel {_\smile }{u}{\!}^{i} } +(\stackrel {_\smile
}{\psi }{\!}^{i} )^{\mathrm {T}} \right )\Delta \stackrel {_\smile }{u}{\!}^{i}
-\sum _{i\in I}\left (\frac {\partial \Phi }{\partial \stackrel {_\frown
}{u}{\!}^{i} } +(\stackrel {_\frown }{\psi }{\!}^{i} )^{\mathrm {T}} \right
)\stackrel {_\frown }{C}{\!}_{i} ^{-1} \stackrel {_\smile }{C}_{i} \Delta
\stackrel {_\smile }{u}{\!}^{i} \\ &\qquad \qquad {}+\sum _{i\in I}\left (\frac
{\partial \Phi }{\partial \stackrel {_\frown }{u}^{i} } +(\stackrel {_\frown
}{\psi }{\!}^{i} )^{\mathrm {T}} \right ) \stackrel {_\frown }{C}_{i} {}^{-1}
\Delta \mathrm {v}^{i} . \end {aligned}$$
Taking this
relation into account in (
A.12), for the increment
of the functional we finally obtain
$$ \begin {aligned} \Delta \mathfrak {I}\thinspace (w,{\rm
v}, \xi )&=\sum _{i\in I}\sum _{k\in I_{i}^{+} } \int \limits _{0}^{l^{ki} }\left
[\frac {\partial f_{0}^{ki} }{\partial u^{ki} } -\left (\frac {d\psi ^{ki}
(x)}{dx} \right )^{\mathrm {T}} -\big (\psi ^{ki} (x)\big )^{T} A^{ki} (x)\right
]\Delta u^{ki} (x)dx \\ &\qquad {}-\sum _{i\in I}\sum _{k\in I_{i}^{+} }\left
[\left (\left . \frac {d\psi ^{ki} (x)}{dx} \right |_{x=\xi ^{ki} } \right
)^{\mathrm {T}} B^{ki} w^{ki} -\frac {\partial \Phi }{\partial \xi ^{ki} } \right
] \Delta \xi ^{ki} \\ &\qquad {}-\sum _{i\in I}\sum _{k\in I_{i}^{+} }\left [\left
(\psi ^{ki} (\xi ^{ki} )\right )^{\mathrm {T}} B^{ki} -\frac {\partial \Phi
}{\partial w^{ki} } \right ] \Delta w^{ki} \\ &\qquad {}+\sum _{i\in I}\left
[\frac {\partial \Phi }{\partial {\rm v}^{i} } +\left (\frac {\partial \Phi
}{\partial \stackrel {_\frown }{u}{\!}^{i} } +(\stackrel {_\frown }{\psi }{\!}^{i}
)^{\mathrm {T}} \right )\stackrel {_\frown }{C}{\!}_{i} ^{-1} \right ] \Delta {\rm
v}^{i} \\ &\qquad {}+\sum _{i\in I}\left [\frac {\partial \Phi }{\partial
\stackrel {_\smile }{u}{\!}^{i} } +(\stackrel {_\smile }{\psi }{\!}^{i} )^{\mathrm
{T}} -\left (\frac {\partial \Phi }{\partial \stackrel {_\frown }{u}{\!}^{i} }
+(\stackrel {_\frown }{\psi }{\!}^{i} )^{\mathrm {T}} \right )\stackrel {_\frown
}{C}{\!}_{i} ^{-1} \stackrel {_\smile }{C}_{i} \right ]\Delta \stackrel {_\smile
}{u}{\!}^{i} +\eta . \end {aligned}$$
(A.16)
Using the arbitrariness of the vector functions \(\psi ^{ki} (x)\in \mathrm {R}^{\aleph } \), \( k\in I_{i}^{k} \), \( i\in I \), we require from them that the expressions in the
brackets (the factors \(\Delta u^{ki} \) and
\(\Delta \!\stackrel {_\smile }{u}{\!}^{i}\)) be zero.
We obtain the boundary value problem (3.5),
(3.6) for the vector functions \(\psi ^{ki}(x) \), \( k\in I_{i}^{+} \), \( i\in I \), which we refer to as the adjoint of problem
(2.1), (2.2). The desired component of the gradient of the functional
\(\mathfrak {I}\thinspace (w,\xi ,{\rm v)}\) will be
determined by the linear parts of the increment of the functional (A.16) based on the increments \(\Delta w^{ki} \), \(\Delta \xi ^{ki} \), and \(\Delta {\rm v}^{i} \) using formulas (3.2), (3.3), and
(3.4).
Thus, Theorem 2 can be
considered to be proved. \(\quad \blacksquare \)
The proof of Theorem 3 follows from the
differentiability of the functional of the problem, the compactness of the admissible sets
\(\Omega _{w^{ki} } \times \Omega _{\xi ^{ki} } \times \Omega _{{\rm v}^{i} } \), \( k\in I_{i}^{+} \), \( i\in I \), the finite dimension of the problem parameter
vector to be optimized, and the necessary optimality conditions in variational form in optimization
problems [10, 11].
Proof of Theorem 4. Let \(\alpha _{j}^{id} (x) \) and \(\gamma _{j}^{i} (x) \) be as yet arbitrary differentiable functions
satisfying (4.5) and condition (4.6). Let us differentiate condition (4.6) taking into account the \((i,d) \)th subsystem of equations in (2.1). After grouping appropriate terms, for \(d\in I_{i}^{-} \) we obtain
$$ \left [\frac {d\alpha _{j}^{id} (x)}{dx} +\alpha
_{j}^{id} (x)A^{id} (x)\right ]u^{id} (x) +\left [-\frac {d\gamma _{j}^{id}
(x)}{dx} +\alpha _{j}^{id} (x)\left (f^{id} (x)+B^{ki} w^{ki} \delta (x-\xi ^{ki}
)\right )\right ]=0.$$
(A.17)
Taking into account the fact that
the functions
\( \alpha _{j}^{id} (x)\) and
\(\gamma _{j}^{i} (x) \) are arbitrary and the necessity of
relation (
A.9) being satisfied for all solutions
\(u^{id} (x) \) of the
\((i,d) \)th subsystem of equations (
2.1), we require that the bracketed expressions be zero. It follows
that
\(\alpha _{j}^{id} (x)\) and
\(\gamma _{j}^{i}(x) \) are the solutions of the Cauchy problems
(
4.5)–(
4.6). The proof of Theorem
4 is complete.
\(\quad \blacksquare \)
