Justification of Direct Scheme for Asymptotic Solving Three-Tempo Linear-Quadratic Control Problems under Weak Nonlinear Perturbations

Abstract

The paper deals with an application of the direct scheme method, consisting of immediately substituting a postulated asymptotic solution into a problem condition and determining a series of control problems for finding asymptotics terms, for asymptotics construction of a solution of a weakly nonlinearly perturbed linear-quadratic optimal control problem with three-tempo state variables. For the first time, explicit formulas for linear-quadratic optimal control problems, from which all terms of the asymptotic expansion are found, are justified, and the estimates of the proximity between the asymptotic and exact solutions are proved for the control, state trajectory, and minimized functional. Non-increasing of the minimized functional, if a next approximation to the optimal control is used, following from the proposed algorithm of the asymptotics construction, is also established.

1. Introduction

Systems with two-tempo variables are the main object in the study of singularly perturbed control problems (see, for instance, the reviews [1,2,3]). However, many practical problems contain multi-tempo fast variables. For instance, such variables arise in models of chain chemical reactions [4], fuel cells with a proton membrane [5], electrical chains [6], electromechanical processes in a synchronous machine [7], power systems [8], nuclear reactors [9], aircraft [10], ocean currents [11], rolling mills [12], two-wheeled carriages [13], forest pests [14], and epidemics [15].

Various asymptotic and numerical (see, for instance, [16]) methods are used for studying singularly perturbed systems with many small parameters standing before derivatives. Basic methods of asymptotic analysis are boundary functions method [17] and integral manifolds method ([18], ch. 7–10), which reduce the considered problem to a problem of simpler structure. The limit passage of an initial problem solution of a system with many small parameters at derivatives, when these parameters tend to zero, was studied for the first time by A.N. Tikhonov [19] and I.S. Gradstein [20]. Asymptotic solution of such problems was first constructed by A.B. Vasil’eva [21].

There are two approaches to constructing asymptotic solutions of optimal control problems. The traditional one is based on an asymptotic solution of a system following from control optimality conditions. Another approach, called the direct scheme method, consists of immediately substituting a postulated asymptotic expansion of a solution into the problem condition and receiving a series of problems for finding asymptotic terms. For two-tempo systems, it is presented, for example, in [22,23]. This method allows for establishing non-increasing of values of the minimized functional if a next optimal control approximation is used. Moreover, standard programs for solving optimal control problems can be applied for finding asymptotics terms. The direct scheme method has been, for instance, used in [24] to obtain any order asymptotic solution of a linear-quadratic optimal control problem with cheap controls of different costs.

The present paper deals with an asymptotic solution construction for the problem $P_{\epsilon }$ with weak nonlinear perturbations in a quadratic performance index and in a linear state equation. Namely, the following functional

$$\begin{array}{c}{J}_{\epsilon }(u)={\int }_0^T(1/2(w{(t,\epsilon )}^{\prime }W(t)w(t,\epsilon )+u{(t,\epsilon )}^{\prime }R(t)u(t,\epsilon ))+\epsilon F(w(t,\epsilon ),u(t,\epsilon ),t,\epsilon ))dt\end{array}$$

(1)

is minimized on trajectories of three-tempo singularly perturbed system

$$\begin{array}{c}\mathcal{E}(\epsilon )\frac{dw(t,\epsilon )}{dt}=A(t)w(t,\epsilon )+B(t)u(t,\epsilon )+\epsilon f(w(t,\epsilon ),u(t,\epsilon ),t,\epsilon ),t\in [0,T],\end{array}$$

(2)

with the initial condition

$$w(0,\epsilon )=w^0.$$

(3)

Here, $\epsilon$ is a non-negative small parameter, $T>0$ is fixed, the prime means transposition; $w(t,\epsilon )={(x{(t,\epsilon )}^{\prime },y{(t,\epsilon )}^{\prime },z{(t,\epsilon )}^{\prime })}^{\prime }$, $x(t,\epsilon )\in {\mathrm{I}\mathrm{R}}^{n_1}$, $y(t,\epsilon )\in {\mathrm{I}\mathrm{R}}^{n_2}$, $z(t,\epsilon )\in {\mathrm{I}\mathrm{R}}^{n_3}$, $u(t,\epsilon )\in {\mathrm{I}\mathrm{R}}^m$; $\mathcal{E}(\epsilon )=diag(I_{n_1},\epsilon I_{n_2},{\epsilon }^2{I}_{n_3})$, $I_{n_i}$ is the identity matrix of order $n_i$, $f=(\stackrel{(1)}{f^{\prime }},\stackrel{(2)}{f^{\prime }},\stackrel{(3)}{f^{\prime }}{)}^{\prime }$, $\stackrel{(i)}{f}\in {\mathrm{I}\mathrm{R}}^{n_i}$, $B=(\stackrel{(1)}{B^{\prime }},\stackrel{(2)}{B^{\prime }},\stackrel{(3)}{B^{\prime }}{)}^{\prime }$, $\stackrel{(i)}{B}:{\mathrm{I}\mathrm{R}}^m\to {\mathrm{I}\mathrm{R}}^{n_i}$, $i=\overline{1,3}$; all functions in (1), (2) are sufficiently smooth with respect to their arguments; for all $t\in [0,T]$ matrices $W(t)$, $R(t)$ are symmetric, moreover, $W(t)$, $R(t)$ and $S(t)=B(t)R{(t)}^{-1}B{(t)}^{\prime }$ are positive definite.

It is assumed that the stability of the matrices $A_{33}$ and $A_{22}-A_{23}{A}_{33}^{-1}{A}_{32}$ takes place. Here, and further $A_{ij}$, $i,j=\overline{1,3}$, mean matrices from a block representation of a matrix A with number of rows and columns $n_1$, $n_2$, $n_3$.

In contrast to [25], where optimal control problems for finding some zero order asymptotics terms for a solution of a nonlinear singularly perturbed problem with three-tempo state variables were formulated, here, explicit expressions of problems for receiving all asymptotic terms are obtained. Note that explicit formulas are very useful for research applying asymptotic methods for solving practical problems.

It should be noted that some results concerning the algorithm of asymptotic solving problem (1)–(3) have been presented in [26], but rigorous proofs and estimates are absent there. Note that [26] deal with matrices in (1), (2) depending on $\epsilon$. However, expanding these matrices with respect to non-negative integer powers of $\epsilon$ and including terms depending on $\epsilon$ into the small nonlinearities, we obtain the problem $P_{\epsilon }$ in our statement.

It is well known that, if a linear-quadratic problem is nonsingular, then its solving is reduced to solving a system of linear differential equations resolved with respect to derivatives. Under studying nonlinear singularly perturbed optimal control problems, it is ordinarily assumed that the control problem is nonsingular, i.e., an optimal control is presented as an explicit function with respect to state and costate variables. See e.g., [27], where, apparently for the first time, singular perturbations methods were used for optimal control problems. In the present paper, unlike these cases, we do not assume the non-singularity of the considered problem for all $\epsilon$ and, for obtaining asymptotic estimates, we analyze a nonlinear singularly perturbed differential-algebraic system.

The essential new results obtained in this paper for problem (1)–(3) are the following:

• The rigorous justification of explicit forms of linear-quadratic optimal control problems, solutions of which are used under constructing an asymptotic solution of nonlinear problem (1)–(3);
• The proof of estimates of the proximity between the exact solution and asymptotic one obtained by the direct scheme method for the control, state trajectory of system (2), (3), and functional (1);
• The proof of non-increasing values of functional (1) under using new asymptotic approximations to the optimal control and constructing minimized sequences.

Throughout the paper, the coefficient with ${\epsilon }^i$ in an expansion of a function $\omega =\omega (\epsilon )$ in a series in powers of $\epsilon$ will be denoted by ${\omega }_i$ or ${[\omega ]}_i$. The k-th partial sum of a series will be denoted by upper wave and the low index k or by braces with the low index k, i.e., ${\tilde{\omega }}_k={\{\omega \}}_k={\sum }_{j=0}^k{\epsilon }^j{\omega }_j$. The functions with negative indices will be considered equal to zero. Positive constants in estimates will be denoted as c and æ.

The paper is organized as follows: in Section 2, we present a formalism of asymptotics construction. Optimal control problems for finding asymptotic terms are given in Section 3. Section 4 is devoted to justification of such a choice of control problems. Namely, transformations of coefficients of expansion of minimized functional with respect to powers of $\epsilon$ with even and odd indices are considered. Asymptotic estimates of the proximity between the asymptotic and exact solutions are proved in Section 5. Non-increasing of the minimized functional, if a next optimal control approximation is used, is also discussed in this section. The last Section 6 contains conclusions.

2. Formalism of Asymptotics Construction

Following the boundary function method by A.B. Vasil’eva (see, for instance, [28]), we will seek a solution of problem (1)–(3) in the form

$$\vartheta (t,\epsilon )=\overline{\vartheta }(t,\epsilon )+{\displaystyle \sum _{i=0}^1}({\mathsf{\Pi }}_i\vartheta ({\tau }_i,\epsilon )+Q_i\vartheta ({\sigma }_i,\epsilon )).$$

(4)

Here, $\vartheta (t,\epsilon )={(w{(t,\epsilon )}^{\prime },u{(t,\epsilon )}^{\prime })}^{\prime }$, $\overline{\vartheta }(t,\epsilon )={\sum }_{j\ge 0}{\epsilon }^j{\overline{\vartheta }}_j(t)$, ${\mathsf{\Pi }}_i\vartheta ({\tau }_i,\epsilon )={\sum }_{j\ge 0}{\epsilon }^j{\mathsf{\Pi }}_{ij}\vartheta ({\tau }_i)$, $Q_i\vartheta ({\sigma }_i,\epsilon )={\sum }_{j\ge 0}{\epsilon }^j{Q}_{ij}\vartheta ({\sigma }_i)$, ${\tau }_i=t/{\epsilon }^{i+1},$ ${\sigma }_i=(t-T)/{\epsilon }^{i+1}$, $i=0,1$, ${\overline{\vartheta }}_j(t)$ are regular functions, ${\mathsf{\Pi }}_{ij}\vartheta ({\tau }_i)$ and $Q_{ij}\vartheta ({\sigma }_i)$ are boundary functions of exponential type in neighborhoods $t=0$ and $t=T$, respectively, i.e.,

$$\begin{array}{c}\parallel {\mathsf{\Pi }}_{ij}\vartheta ({\tau }_i)\parallel ⩽cexp(-\ae {\tau }_i),{\tau }_i⩾0,\parallel Q_{ij}\vartheta ({\sigma }_i)\parallel ⩽cexp(\ae {\sigma }_i),{\sigma }_i⩽0,\end{array}$$

where c and æ are positive constants independent of the arguments of functions under study.

For any sufficiently smooth function $G(w(t,\epsilon ),u(t,\epsilon ),t,\epsilon )$, we will use the notation $G(\vartheta (t,\epsilon ),t,\epsilon )$ and the asymptotic representation

$$G(\vartheta ,t,\epsilon )=\overline{G}(t,\epsilon )+{\displaystyle \sum _{i=0}^1}({\mathsf{\Pi }}_{i}G({\tau }_i,\epsilon )+Q_{i}G({\sigma }_i,\epsilon )),$$

(5)

$$\begin{gathered} \overline{G}(t,\epsilon )=G(\overline{\vartheta }(t,\epsilon ),t,\epsilon )={\sum }_{j\ge 0}{\epsilon }^j{\overline{G}}_j(t),{\mathsf{\Pi }}_{0}G({\tau }_0,\epsilon )=G(\overline{\vartheta }(\epsilon {\tau }_0,\epsilon )+{\mathsf{\Pi }}_0\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon ) \\ -G(\overline{\vartheta }(\epsilon {\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon )={\sum }_{j\ge 0}{\epsilon }^j{\mathsf{\Pi }}_{0j}G({\tau }_0),{\mathsf{\Pi }}_{1}G({\tau }_1,\epsilon )=G(\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon ) \\ +{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )-G(\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )={\sum }_{j\ge 0}{\epsilon }^j{\mathsf{\Pi }}_{1j}G({\tau }_1), \\ {Q}_{0}G({\sigma }_0,\epsilon )=G(\overline{\vartheta }(T+\epsilon {\sigma }_0,\epsilon )+Q_0\vartheta ({\sigma }_0,\epsilon ),T+\epsilon {\sigma }_0,\epsilon )-G(\overline{\vartheta }(T+\epsilon {\sigma }_0,\epsilon ),T+\epsilon {\sigma }_0,\epsilon ) \\ ={\sum }_{j\ge 0}{\epsilon }^j{Q}_{0j}G({\sigma }_0)Q_{1}G({\sigma }_1,\epsilon )=G(\overline{\vartheta }(T+{\epsilon }^2{\sigma }_1,\epsilon )+Q_0\vartheta (\epsilon {\sigma }_1,\epsilon )+Q_1\vartheta ({\sigma }_1,\epsilon ),T+{\epsilon }^2{\sigma }_1,\epsilon ) \\ -G(\overline{\vartheta }(T+{\epsilon }^2{\sigma }_1,\epsilon )+Q_0\vartheta (\epsilon {\sigma }_1,\epsilon ),T+{\epsilon }^2{\sigma }_1,\epsilon )={\sum }_{j\ge 0}{\epsilon }^j{Q}_{1j}G({\sigma }_1). \end{gathered}$$

Substitute (4) in (1) and present the integrand in the form of sum (4). Passing in the integrals from the expressions depending on ${\tau }_i$, ${\sigma }_i$, $i=0,1$, to integrals over the corresponding intervals $[0,+\infty )$ and $(-\infty ,0]$, we obtain the following expansion of the functional (1)

$$J_{\epsilon }(u)=\sum _{j\ge 0}{\epsilon }^j{J}_j.$$

(6)

Substituting expansion (4) into system (2) and initial value (3), using (5), then equating terms of the same powers of $\epsilon$, separately depending on regular and different boundary functions, we obtain relations for defining asymptotics terms.

Introducing the notation $E_1=diag(I_{n_1},0,0)$, $E_2=diag(0,I_{n_2},0)$, $E_3=diag(0,0,I_{n_3})$, and $\varphi (\vartheta ,t,\epsilon )=A(t)w(t,\epsilon )+B(t)u(t,\epsilon )+\epsilon f(w(t,\epsilon ),u(t,\epsilon ),t,\epsilon )$, we obtain the following equations:

$$\begin{array}{c}{E}_1\frac{d{\overline{w}}_j(t)}{dt}+E_2\frac{d{\overline{w}}_{j-1}(t)}{dt}+E_3\frac{d{\overline{w}}_{j-2}(t)}{dt}={[\overline{\varphi }(t,\epsilon )]}_j,\end{array}$$

(7)

$$\begin{array}{c}(E_1+E_2)\frac{d{\mathsf{\Pi }}_{0j}w({\tau }_0)}{d{\tau }_0}+E_3\frac{d{\mathsf{\Pi }}_{0(j-1)}w({\tau }_0)}{d{\tau }_0}=E_1{[{\mathsf{\Pi }}_0\varphi ({\tau }_0,\epsilon )]}_{j-1}\\ +(E_2+E_3){[{\mathsf{\Pi }}_0\varphi ({\tau }_0,\epsilon )]}_j,\end{array}$$

(8)

$$\begin{array}{c}(E_1+E_2)\frac{d{Q}_{0j}w({\sigma }_0)}{d{\sigma }_0}+E_3\frac{d{Q}_{0(j-1)}w({\sigma }_0)}{d{\sigma }_0}=E_1{[Q_0\varphi ({\sigma }_0,\epsilon )]}_{j-1}\\ +(E_2+E_3){[Q_0\varphi ({\sigma }_0,\epsilon )]}_j,\end{array}$$

(9)

$$\begin{array}{c}\frac{d{\mathsf{\Pi }}_{1j}w({\tau }_1)}{d{\tau }_1}=E_1{[{\mathsf{\Pi }}_1\varphi ({\tau }_1,\epsilon )]}_{j-2}+E_2{[{\mathsf{\Pi }}_1\varphi ({\tau }_1,\epsilon )]}_{j-1}+E_3{[{\mathsf{\Pi }}_1\varphi ({\tau }_1,\epsilon )]}_j,\end{array}$$

(10)

$$\begin{array}{c}\frac{d{Q}_{1j}w({\sigma }_1)}{d{\sigma }_1}=E_1{[Q_1\varphi ({\sigma }_1,\epsilon )]}_{j-2}+E_2{[Q_1\varphi ({\sigma }_1,\epsilon )]}_{j-1}+E_3{[Q_1\varphi ({\sigma }_1,\epsilon )]}_j.\end{array}$$

(11)

From Equations (8)–(11) at $j=0$, (10) and (11) at $j=1$, we found the corresponding boundary functions

$$\begin{array}{c}{E}_1{\mathsf{\Pi }}_{00}w({\tau }_0)=0,E_1{\mathsf{\Pi }}_{10}w({\tau }_1)=E_1{\mathsf{\Pi }}_{11}w({\tau }_1)=0,E_1{Q}_{00}w({\sigma }_0)=0,\\ E_1{Q}_{10}w({\sigma }_1)=E_1{Q}_{11}w({\sigma }_1)=0,E_2{\mathsf{\Pi }}_{10}w({\tau }_1)=0,E_2{Q}_{10}w({\sigma }_1)=0.\end{array}$$

(12)

In view of the last equalities, from (3), we obtain relations for initial values

$$E_1{\overline{w}}_0(0)=E_1{w}^0,E_1({\overline{w}}_1(0)+{\mathsf{\Pi }}_{01}w(0))=0,$$

(13)

$$E_1({\overline{w}}_j(0)+{\mathsf{\Pi }}_{0j}w(0)+{\mathsf{\Pi }}_{1j}w(0))=0,j\ge 2,$$

(14)

$$E_2({\overline{w}}_0(0)+{\mathsf{\Pi }}_{00}w(0))=E_2{w}^0,$$

(15)

$$E_2({\overline{w}}_j(0)+{\mathsf{\Pi }}_{0j}w(0)+{\mathsf{\Pi }}_{1j}w(0))=0,j\ge 1,$$

(16)

$$E_3({\overline{w}}_j(0)+{\mathsf{\Pi }}_{0j}w(0)+{\mathsf{\Pi }}_{1j}w(0))=\left\{\begin{array}{c}{E}_3{w}^0,j=0,\hfill \\ 0,j\ge 1.\hfill \end{array}\right.$$

(17)

Remark 1. 

If boundary functions ${\mathsf{\Pi }}_{ij}w$, $Q_{ij}w$, $i=0,1$, $j=\overline{0,n-1}$ have been found, then, from Equations (8)–(11), it follows the corollary that functions $E_1{\mathsf{\Pi }}_{in}w({\tau }_i)$, $E_1{Q}_{in}w({\sigma }_i)$, $i=0,1$, $E_2{\mathsf{\Pi }}_{1n}w({\tau }_1)$, $E_2{Q}_{1n}w({\sigma }_1)$, and $E_1{\mathsf{\Pi }}_{1(n+1)}w({\tau }_1)$, $E_1{Q}_{1(n+1)}w({\sigma }_1)$ are known.

3. Optimal Control Problems for Finding Asymptotics Terms

In this section, forms of control problems for finding asymptotics terms will be given. In contrast to [26], the justification of these relations will be presented.

With the help of the notations,

$$\begin{array}{c}\hfill \rho (\vartheta ,\psi ,t,\epsilon )=W(t)w(t,\epsilon )-A{(t)}^{\prime }\psi (t,\epsilon )+\epsilon (F_w{(\vartheta ,t,\epsilon )}^{\prime }-f_w{(\vartheta ,t,\epsilon )}^{\prime }\psi (t,\epsilon )),\\ \hfill \chi (\vartheta ,\psi ,t,\epsilon )=R(t)u(t,\epsilon )-B{(t)}^{\prime }\psi (t,\epsilon )+\epsilon (F_u{(\vartheta ,t,\epsilon )}^{\prime }-f_u{(\vartheta ,t,\epsilon )}^{\prime }\psi (t,\epsilon )),\end{array}$$

five optimal control problems ${\overline{P}}_j$, ${\mathsf{\Pi }}_{ij}P$, $Q_{ij}P$, $i=0,1$, for determining asymptotics terms in expansion (4) will be written. Costate variables in these problems will be denoted as ${\overline{\psi }}_j(t)$, ${\mathsf{\Pi }}_{ij}\psi ({\tau }_i)$, $Q_{ij}\psi ({\sigma }_i)$, $i=0,1$, respectively.

Furthermore, the hat and the low index k in a function notation will mean that the function is calculated with the functional argument equal to the k-th partial sum of the corresponding expansion, e.g., ${\widehat{\overline{f}}}_k(t,\epsilon )=f({\tilde{\overline{\vartheta }}}_k(t,\epsilon ),t,\epsilon )$.

In the following expressions with $\rho$ and $\chi$ in the performance indices of the formulated optimal control problems, we take $\psi (t,\epsilon )={\sum }_{j=0}^{\infty }{\epsilon }^j({\overline{\psi }}_j(t)+(\epsilon E_1+E_2+E_3)({\mathsf{\Pi }}_{0j}\psi ({\tau }_0)+Q_{0j}\psi ({\sigma }_0))+({\epsilon }^2{E}_1+\epsilon E_2+E_3)({\mathsf{\Pi }}_{1j}\psi ({\tau }_1)+Q_{1j}\psi ({\sigma }_1)))$.

Regular functions ${\overline{\vartheta }}_j(t)$, $t\in [0,T]$, are determined as solutions of problems ${\overline{P}}_j$, which consist of minimizing the functional

$$\begin{array}{c}{\overline{J}}_j({\overline{u}}_j)={\overline{w}}_j{(T)}^{\prime }{E}_1(Q_{0(j-1)}\psi (0)+Q_{1(j-2)}\psi (0))+\underset{0}{\overset{T}{\int }}({\overline{w}}_j{(t)}^{\prime }(\frac{1}{2}W(t){\overline{w}}_j(t)\\ +{[{\widehat{\overline{\rho }}}_{j-1}(t,\epsilon )]}_j-E_2\frac{d{\overline{\psi }}_{j-1}(t)}{dt}-E_3\frac{d{\overline{\psi }}_{j-2}(t)}{dt})+{\overline{u}}_j{(t)}^{\prime }(\frac{1}{2}R(t){\overline{u}}_j(t)+{[{\widehat{\overline{\chi }}}_{j-1}(t,\epsilon )]}_j))dt\end{array}$$

on trajectories of system (7) with initial conditions from (13) or (14) in dependence on j.

The boundary functions ${\mathsf{\Pi }}_{0j}\vartheta ({\tau }_0)$, ${\tau }_0\in [0,+\infty )$ are determined from optimal control problems ${\mathsf{\Pi }}_{0j}P$ consisting of minimizing the functional

$$\begin{array}{c}{\mathsf{\Pi }}_{0j}J({\mathsf{\Pi }}_{0j}u)=\underset{0}{\overset{+\infty }{\int }}({\mathsf{\Pi }}_{0j}w{({\tau }_0)}^{\prime }(\frac{1}{2}W(0){\mathsf{\Pi }}_{0j}w({\tau }_0)+{[{\widehat{\mathsf{\Pi }}}_{0(j-1)}\rho ({\tau }_0,\epsilon )]}_j-E_3\frac{d{\mathsf{\Pi }}_{0(j-1)}\psi ({\tau }_0)}{d{\tau }_0})\\ +{\mathsf{\Pi }}_{0j}u{({\tau }_0)}^{\prime }(\frac{1}{2}R(0){\mathsf{\Pi }}_{0j}u({\tau }_0)+{[{\widehat{\mathsf{\Pi }}}_{0(j-1)}\chi ({\tau }_0,\epsilon )]}_j))d{\tau }_0\end{array}$$

on trajectories of system (8) with the conditions ${\mathsf{\Pi }}_{0j}x(+\infty )=0$ and (15) or (16) in dependence on j.

The boundary functions $Q_{0j}\vartheta ({\sigma }_0)$, ${\sigma }_0\in (-\infty ,0]$, are determined from optimal control problems $Q_{0j}P$ consisting of minimizing the functional

$$\begin{array}{c}{Q}_{0j}J(Q_{0j}u)=Q_{0j}w{(0)}^{\prime }{E}_2({\overline{\psi }}_j(T)+Q_{1(j-1)}\psi (0))\\ +\underset{-\infty }{\overset{0}{\int }}(Q_{0j}w{({\sigma }_0)}^{\prime }(\frac{1}{2}W(T)Q_{0j}w({\sigma }_0)+{[{\widehat{Q}}_{0(j-1)}\rho ({\sigma }_0,\epsilon )]}_j-E_3\frac{d{Q}_{0(j-1)}\psi ({\sigma }_0)}{d{\sigma }_0})\\ +Q_{0j}u{({\sigma }_0)}^{\prime }(\frac{1}{2}R(T)Q_{0j}u({\sigma }_0)+{[{\widehat{Q}}_{0(j-1)}\chi ({\sigma }_0,\epsilon )]}_j))d{\sigma }_0\end{array}$$

on trajectories of system (9) with the condition $(E_1+E_2)Q_{0j}w(-\infty )=0$.

The boundary functions ${\mathsf{\Pi }}_{1j}\vartheta ({\tau }_1)$, ${\tau }_1\in [0,+\infty )$, are determined from optimal control problems ${\mathsf{\Pi }}_{1j}P$ consisting of minimizing the functional

$$\begin{array}{c}{\mathsf{\Pi }}_{1j}J({\mathsf{\Pi }}_{1j}u)=\underset{0}{\overset{+\infty }{\int }}({\mathsf{\Pi }}_{1j}w{({\tau }_1)}^{\prime }(\frac{1}{2}W(0){\mathsf{\Pi }}_{1j}w({\tau }_1)+{[{\widehat{\mathsf{\Pi }}}_{1(j-1)}\rho ({\tau }_1,\epsilon )]}_j)\\ +{\mathsf{\Pi }}_{1j}u{({\tau }_1)}^{\prime }(\frac{1}{2}R(0){\mathsf{\Pi }}_{1j}u({\tau }_1)+{[{\widehat{\mathsf{\Pi }}}_{1(j-1)}\chi ({\tau }_1,\epsilon )]}_j))d{\tau }_1\end{array}$$

on trajectories of system (10) with the conditions $(E_1+E_2){\mathsf{\Pi }}_{1j}w(+\infty )=0$ and (17).

The boundary functions $Q_{1j}\vartheta ({\sigma }_1)$, ${\sigma }_1\in (-\infty ,0]$, are determined from optimal control problems $Q_{1j}P$ consisting of minimizing the functional

$$\begin{array}{c}{Q}_{1j}J(Q_{1j}u)=Q_{1j}w{(0)}^{\prime }{E}_3({\overline{\psi }}_j(T)+Q_{0j}\psi (0))+\underset{-\infty }{\overset{0}{\int }}(Q_{1j}w{({\sigma }_1)}^{\prime }(\frac{1}{2}W(T)Q_{1j}w({\sigma }_1)\end{array}$$

$$\begin{array}{c}+{[{\widehat{Q}}_{1(j-1)}\rho ({\sigma }_1,\epsilon )]}_j)+Q_{1j}u{({\sigma }_1)}^{\prime }(\frac{1}{2}R(T)Q_{1j}u({\sigma }_1)+{[{\widehat{Q}}_{1(j-1)}\chi ({\sigma }_1,\epsilon )]}_j))d{\sigma }_1\end{array}$$

on trajectories of system (11) with the condition $Q_{1j}w(-\infty )=0$.

Remark 2. 

Though the original problem (1)–(3) is nonlinear, the considered optimal control problems ${\overline{P}}_j$, ${\mathsf{\Pi }}_{ij}P$, $Q_{ij}P$, $i=0,1$, are linear-quadratic.

Solutions of the formulated optimal control problems can be found from the control optimality conditions in the Pontryagin maximum principle form. Namely, a solution of the problem ${\overline{P}}_j$ can be found from (7), (13), or (14) in dependence on j, and the relations

$$B{(t)}^{\prime }{\overline{\psi }}_j(t)-R(t){\overline{u}}_j(t)-{[{\widehat{\overline{\chi }}}_{j-1}(t,\epsilon )]}_j=0,$$

(18)

$$E_1\frac{d{\overline{\psi }}_j(t)}{dt}=W(t){\overline{w}}_j(t)-A{(t)}^{\prime }{\overline{\psi }}_j(t)+{[{\widehat{\overline{\rho }}}_{j-1}(t,\epsilon )]}_j-E_2\frac{d{\overline{\psi }}_{j-1}(t)}{dt}-E_3\frac{d{\overline{\psi }}_{j-2}(t)}{dt},$$

(19)

$$E_1{\overline{\psi }}_j(T)=-E_1(Q_{0(j-1)}\psi (0)+Q_{1(j-2)}\psi (0)).$$

(20)

A solution of the problem ${\mathsf{\Pi }}_{0j}P$ with $E_1{\mathsf{\Pi }}_{0j}w(+\infty )=0$ can be found from (8), (12) and (15) or (16) in dependence on j, and the relations

$$B{(0)}^{\prime }(E_2+E_3){\mathsf{\Pi }}_{0j}\psi -R(0){\mathsf{\Pi }}_{0j}u-{[{\widehat{\mathsf{\Pi }}}_{0(j-1)}\chi ({\tau }_0,\epsilon )]}_j=0,$$

(21)

$$\begin{array}{c}(E_1+E_2)\frac{d{\mathsf{\Pi }}_{0j}\psi }{d{\tau }_0}=W(0){\mathsf{\Pi }}_{0j}w-A{(0)}^{\prime }(E_2+E_3){\mathsf{\Pi }}_{0j}\psi \\ +{[{\widehat{\mathsf{\Pi }}}_{0(j-1)}\rho ({\tau }_0,\epsilon )]}_j-E_3\frac{d{\mathsf{\Pi }}_{0(j-1)}\psi }{d{\tau }_0},\end{array}$$

(22)

$$(E_1+E_2){\mathsf{\Pi }}_{0j}\psi (+\infty )=0.$$

A solution of the problem $Q_{0j}P$ with $(E_1+E_2)Q_{0j}w(-\infty )=0$ can be found from (9), (12) and the relations

$$B{(T)}^{\prime }(E_2+E_3)Q_{0j}\psi -R(T)Q_{0j}u-{[{\widehat{Q}}_{0(j-1)}\chi ({\sigma }_0,\epsilon )]}_j=0,$$

$$\begin{array}{c}(E_1+E_2)\frac{d{Q}_{0j}\psi }{d{\sigma }_0}=W(T)Q_{0j}w-A{(T)}^{\prime }(E_2+E_3)Q_{0j}\psi \\ +{[{\widehat{Q}}_{0(j-1)}\rho ({\sigma }_0,\epsilon )]}_j-E_3\frac{d{Q}_{0(j-1)}\psi }{d{\sigma }_0},\end{array}$$

$$E_1{Q}_{0j}\psi (-\infty )=0,E_2{Q}_{0j}\psi (0)=-E_2({\overline{\psi }}_j(T)+Q_{1(j-1)}\psi (0)).$$

(23)

A solution of the problem ${\mathsf{\Pi }}_{1j}P$ with $(E_1+E_2){\mathsf{\Pi }}_{1j}w(+\infty )=0$ can be found from (10), (12), (17) in dependence on j, and the relations

$$B{(0)}^{\prime }{E}_3{\mathsf{\Pi }}_{1j}\psi -R(0){\mathsf{\Pi }}_{1j}u-{[{\widehat{\mathsf{\Pi }}}_{1(j-1)}\chi ({\tau }_1,\epsilon )]}_j=0,$$

(24)

$$\begin{array}{c}\frac{d{\mathsf{\Pi }}_{1j}\psi }{d{\tau }_1}=W(0){\mathsf{\Pi }}_{1j}w-A{(0)}^{\prime }{E}_3{\mathsf{\Pi }}_{1j}\psi +{[{\widehat{\mathsf{\Pi }}}_{1(j-1)}\rho ({\tau }_1,\epsilon )]}_j,\end{array}$$

(25)

$${\mathsf{\Pi }}_{1j}\psi (+\infty )=0.$$

A solution of the problem $Q_{1j}P$ with $Q_{1j}w(-\infty )=0$ can be found from (11), (12) in dependence on j, and the relations

$$B{(T)}^{\prime }{E}_3{Q}_{1j}\psi -R(T)Q_{1j}u-{[{\widehat{Q}}_{1(j-1)}\chi ({\sigma }_1,\epsilon )]}_j=0,$$

$$\begin{array}{c}\frac{d{Q}_{1j}\psi }{d{\sigma }_1}=W(T)Q_{1j}w-A{(T)}^{\prime }{E}_3{Q}_{1j}\psi +{[{\widehat{Q}}_{1(j-1)}\rho ({\sigma }_1,\epsilon )]}_j,\end{array}$$

$$(E_1+E_2)Q_{1j}\psi (-\infty )=0,E_3{Q}_{1j}\psi (0)=-E_3({\overline{\psi }}_j(T)+Q_{0j}\psi (0)).$$

(26)

In view of the control optimality condition in the Pontryagin maximum principle, a solution of the problem (1)–(3) satisfies (2), (3) and the following relations, including the costate variable $\phi (t,\epsilon )={(\zeta {(t,\epsilon )}^{\prime },\eta {(t,\epsilon )}^{\prime },\theta {(t,\epsilon )}^{\prime })}^{\prime }$,

$$B{(t)}^{\prime }\phi -R(t)u-\epsilon (F_u{(\vartheta ,t,\epsilon )}^{\prime }-f_u{(\vartheta ,t,\epsilon )}^{\prime }\phi )=0,$$

(27)

$$\begin{array}{c}\hfill \mathcal{E}(\epsilon )\frac{d\phi }{dt}=W(t)w-A{(t)}^{\prime }\phi +\epsilon (F_w{(\vartheta ,t,\epsilon )}^{\prime }-f_w{(\vartheta ,t,\epsilon )}^{\prime }\phi ),\end{array}$$

(28)

$$\phi (T,\epsilon )=0.$$

(29)

An asymptotic solution of problems (2), (3), (27)–(29) can be constructed in the form (4), i.e., in addition, we set

$$\phi (t,\epsilon )=\overline{\phi }(t,\epsilon )+{\displaystyle \sum _{i=0}^1}({\mathsf{\Pi }}_i\phi ({\tau }_i,\epsilon )+Q_i\phi ({\sigma }_i,\epsilon )),$$

(30)

where all terms have the properties of the corresponding terms in (4).

Substitute asymptotic expansions (4), (30) into (27)–(29) and use presentation (5). Introducing the notation $g(\vartheta ,\phi ,t,\epsilon )=\rho (\vartheta ,\phi ,t,\epsilon )$, $h(\vartheta ,\phi ,t,\epsilon )=\chi (\vartheta ,\phi ,t,\epsilon )$ and equating terms of the same power of $\epsilon$ separately depending on t, ${\tau }_i$, ${\sigma }_i$, $i=0,1$, we obtain the relations

$$B{(t)}^{\prime }{\overline{\phi }}_j-R(t){\overline{u}}_j-{[{\widehat{\overline{h}}}_{j-1}(t,\epsilon )]}_j=0,$$

$$E_1\frac{d{\overline{\phi }}_j}{dt}+E_2\frac{d{\overline{\phi }}_{j-1}}{dt}+E_3\frac{d{\overline{\phi }}_{j-2}}{dt}=W(t){\overline{w}}_j-A{(t)}^{\prime }{\overline{\phi }}_j+{[{\widehat{\overline{g}}}_{j-1}(t,\epsilon )]}_j,$$

$$B{(0)}^{\prime }{\mathsf{\Pi }}_{ij}\phi -R(0){\mathsf{\Pi }}_{ij}u-{[{\widehat{\mathsf{\Pi }}}_{i(j-1)}h({\tau }_i,\epsilon )]}_j=0,$$

$$\begin{array}{c}{E}_1\frac{d{\mathsf{\Pi }}_{ij}\phi }{d{\tau }_i}+E_2\frac{d{\mathsf{\Pi }}_{i(j-1)}\phi }{d{\tau }_i}+E_3\frac{d{\mathsf{\Pi }}_{i(j-2)}\phi }{d{\tau }_i}=W(0){\mathsf{\Pi }}_{i(j-i-1)}w\\ -A{(0)}^{\prime }{\mathsf{\Pi }}_{i(j-i-1)}\phi +{[{\widehat{\mathsf{\Pi }}}_{i(j-i-2)}g({\tau }_i,\epsilon )]}_{j-i-1},\end{array}$$

(31)

$$B{(T)}^{\prime }{Q}_{ij}\phi -R(T)Q_{ij}u-{[{\widehat{Q}}_{i(j-1)}h({\sigma }_i,\epsilon )]}_j=0,$$

$$\begin{array}{c}{E}_1\frac{d{Q}_{ij}\phi }{d{\sigma }_i}+E_2\frac{d{Q}_{i(j-1)}\phi }{d{\sigma }_i}+E_3\frac{d{Q}_{i(j-2)}\phi }{d{\sigma }_i}=W(T)Q_{i(j-i-1)}w\\ -A{(T)}^{\prime }{Q}_{i(j-i-1)}\phi +{[{\widehat{Q}}_{i(j-i-2)}g({\sigma }_i,\epsilon )]}_{j-i-1},\end{array}$$

(32)

$${\overline{\phi }}_j(T)+Q_{0j}\phi (0)+Q_{1j}\phi (0)=0.$$

(33)

It follows from (31), (32) with $j=0$ and $i=j=1$ that

$$\begin{array}{c}{E}_1{\mathsf{\Pi }}_{00}\phi ({\tau }_0)=0,E_1{\mathsf{\Pi }}_{10}\phi ({\tau }_1)=E_1{\mathsf{\Pi }}_{11}\phi ({\tau }_1)=0,E_1{Q}_{00}\phi ({\sigma }_0)=0,\\ E_1{Q}_{10}\phi ({\sigma }_1)=E_1{Q}_{11}\phi ({\sigma }_1)=0,E_2{\mathsf{\Pi }}_{10}\phi ({\tau }_1)=0,E_2{Q}_{10}\phi ({\sigma }_1)=0.\end{array}$$

4. Justification of Formalism of Asymptotics Construction

This section deals with the establishment of a relation between the forms of coefficients in the expansion (6) of the minimized functional with respect to powers of $\epsilon$ and the expressions of the performance indices in optimal control problems formulated in the previous section. The following theorem, which was given in [26] without any rigorous proof, will be further justified.

Theorem 1. 

The sum ${\overline{J}}_j+{\mathsf{\Pi }}_{1(j-1)}J+Q_{1(j-1)}J$ of the performance indices in problems ${\overline{P}}_j$, ${\mathsf{\Pi }}_{1(j-1)}P$, $Q_{1(j-1)}P$ is obtained by transforming the coefficient $J_{2j}$ in expansion (6) and dropping terms, which are known after solving problems ${\overline{P}}_k$, ${\mathsf{\Pi }}_{0k}P$, $Q_{0k}P$, $k=\overline{0,j-1}$, ${\mathsf{\Pi }}_{1k}P$, $Q_{1k}P$, $k=\overline{0,k-2}$. The sum ${\mathsf{\Pi }}_{0j}J+Q_{0j}J$ of the performance indices in problems ${\mathsf{\Pi }}_{0j}P$, $Q_{0j}P$ is obtained by transforming the coefficient $J_{2j+1}$ in expansion (6) and dropping terms, which are known after solving problems ${\overline{P}}_k$, $k=\overline{0,j}$, ${\mathsf{\Pi }}_{ik}P$, $Q_{ik}P$, $i=0,1$, $k=\overline{0,j-1}$.

Proof. 

Denote the integrand in (1) by means $\mathbb{F}(\vartheta ,t,\epsilon )$. In view of (5), we can present $J_k$ in the form

$$\begin{array}{c}{J}_k={\int }_0^T{\overline{\mathbb{F}}}_k(t)dt+{\int }_0^{+\infty }{\mathsf{\Pi }}_{0(k-1)}\mathbb{F}({\tau }_0)d{\tau }_0\\ +{\int }_{-\infty }^0{Q}_{0(k-1)}\mathbb{F}({\sigma }_0)d{\sigma }_0+{\int }_0^{+\infty }{\mathsf{\Pi }}_{1(k-2)}\mathbb{F}({\tau }_1)d{\tau }_1+{\int }_{-\infty }^0{Q}_{1(k-2)}\mathbb{F}({\sigma }_1)d{\sigma }_1.\end{array}$$

(34)

It is clear that the last expression contains the asymptotics terms with numbers more than it is necessary in this theorem, for instance, ${\overline{\mathbb{F}}}_{2n}(t)={[\mathbb{F}({\tilde{\overline{\vartheta }}}_{2n}(t,\epsilon ),t,\epsilon )]}_{2n}$. In order to prove the theorem, we will use control optimality conditions for formulated previously control problems.

It is evident that the coefficient $J_0$ in (6) is the performance index in problem ${\overline{P}}_0$.

We will analyze the coefficient $J_1$. In view of (34) with $k=1$, we have

$$\begin{array}{c}{J}_1={\int }_0^T{\overline{\mathbb{F}}}_1(t)dt+{\int }_0^{+\infty }{\mathsf{\Pi }}_{00}\mathbb{F}({\tau }_0)d{\tau }_0+{\int }_{-\infty }^0{Q}_{00}\mathbb{F}({\sigma }_0)d{\sigma }_0\\ ={\int }_0^T({\overline{w}}_1{(t)}^{\prime }W(t){\overline{w}}_0(t)+{\overline{u}}_1{(t)}^{\prime }R(t){\overline{u}}_0(t)+{[{\widehat{\overline{\mathbb{F}}}}_0(t,\epsilon )]}_1)dt\\ +{\int }_0^{+\infty }(\frac{1}{2}({\mathsf{\Pi }}_{00}w{({\tau }_0)}^{\prime }W(0){\mathsf{\Pi }}_{00}w({\tau }_0)+{\mathsf{\Pi }}_{00}u{({\tau }_0)}^{\prime }R(0){\mathsf{\Pi }}_{00}u({\tau }_0))\\ +{\mathsf{\Pi }}_{00}w{({\tau }_0)}^{\prime }W(0){\overline{w}}_0(0)+{\mathsf{\Pi }}_{00}u{({\tau }_0)}^{\prime }R(0){\overline{u}}_0(0))d{\tau }_0\\ +{\int }_{-\infty }^0(\frac{1}{2}(Q_{00}w{({\sigma }_0)}^{\prime }W(T)Q_{00}w({\sigma }_0)+Q_{00}u{({\sigma }_0)}^{\prime }R(T)Q_{00}u({\sigma }_0))\\ +Q_{00}w{({\sigma }_0)}^{\prime }W(T){\overline{w}}_0(T)+Q_{00}u{({\sigma }_0)}^{\prime }R(T){\overline{u}}_0(T))d{\sigma }_0.\end{array}$$

Transforming the following expression from $J_1$ with the help of control optimality conditions for the problem ${\overline{P}}_0$ (see (18)–(20) with $j=0$), the integration by parts, and also (12), (7) with $j=1$, (8), (9) with $j=0$ and $j=1$, and (15), we have

$$\begin{array}{c}{\int }_0^T({\overline{w}}_1{(t)}^{\prime }W(t){\overline{w}}_0(t)+{\overline{u}}_1{(t)}^{\prime }R(t){\overline{u}}_0(t))dt+{\int }_0^{+\infty }({\mathsf{\Pi }}_{00}w{({\tau }_0)}^{\prime }W(0){\overline{w}}_0(0)\\ +{\mathsf{\Pi }}_{00}u{({\tau }_0)}^{\prime }R(0){\overline{u}}_0(0))d{\tau }_0+{\int }_{-\infty }^0(Q_{00}w{({\sigma }_0)}^{\prime }W(T){\overline{w}}_0(T)+Q_{00}u{({\sigma }_0)}^{\prime }R(T){\overline{u}}_0(T))d{\sigma }_0\\ ={\int }_0^T({\overline{w}}_1{(t)}^{\prime }(E_1\frac{d{\overline{\psi }}_0(t)}{dt}+A{(t)}^{\prime }{\overline{\psi }}_0(t))+{\overline{u}}_1{(t)}^{\prime }B{(t)}^{\prime }{\overline{\psi }}_0(t))dt\\ +{\int }_0^{+\infty }({\mathsf{\Pi }}_{00}w{({\tau }_0)}^{\prime }(E_1\frac{d{\overline{\psi }}_0}{dt}(0)+A{(0)}^{\prime }{\overline{\psi }}_0(0))+{\mathsf{\Pi }}_{00}u{({\tau }_0)}^{\prime }B{(0)}^{\prime }{\overline{\psi }}_0(0))d{\tau }_0\\ +{\int }_{-\infty }^0(Q_{00}w{({\sigma }_0)}^{\prime }(E_1\frac{d{\overline{\psi }}_0}{dt}(T)+A{(T)}^{\prime }{\overline{\psi }}_0(T))+Q_{00}u{({\sigma }_0)}^{\prime }B{(T)}^{\prime }{\overline{\psi }}_0(T))d{\sigma }_0\\ ={\overline{w}}_1{(t)}^{\prime }{E}_1{\overline{\psi }}_0{(t)|}_0^T+{\int }_0^T{\overline{\psi }}_0{(t)}^{\prime }(-E_1\frac{d{\overline{w}}_1}{dt}(t)+A(t){\overline{w}}_1(t)+B(t){\overline{u}}_1(t))dt\\ +{\int }_0^{+\infty }{\overline{\psi }}_0{(0)}^{\prime }(A(0){\mathsf{\Pi }}_{00}w({\tau }_0)+B(0){\mathsf{\Pi }}_{00}u({\tau }_0))d{\tau }_0\\ +{\int }_{-\infty }^0{\overline{\psi }}_0{(T)}^{\prime }(A(T)Q_{00}w({\sigma }_0)+B(T)Q_{00}u({\sigma }_0))d{\sigma }_0\\ ={\mathsf{\Pi }}_{01}w{(0)}^{\prime }{E}_1{\overline{\psi }}_0(0)+{\int }_0^T{\overline{\psi }}_0{(t)}^{\prime }(E_2\frac{d{\overline{w}}_0}{dt}(t)-{[{\widehat{\overline{\varphi }}}_0(t,\epsilon )]}_1)dt\end{array}$$

$$\begin{array}{c}+{\int }_0^{+\infty }{\overline{\psi }}_0{(0)}^{\prime }(E_1\frac{d{\mathsf{\Pi }}_{01}w({\tau }_0)}{d{\tau }_0}+E_2\frac{d{\mathsf{\Pi }}_{00}w({\tau }_0)}{d{\tau }_0})d{\tau }_0\\ +{\int }_{-\infty }^0{\overline{\psi }}_0{(T)}^{\prime }(E_1\frac{d{Q}_{01}w({\sigma }_0)}{d{\sigma }_0}+E_2\frac{d{Q}_{00}w({\sigma }_0)}{d{\sigma }_0})d{\sigma }_0\\ ={\mathsf{\Pi }}_{01}w{(0)}^{\prime }{E}_1{\overline{\psi }}_0(0)+{\int }_0^T{\overline{\psi }}_0{(t)}^{\prime }(E_2\frac{d{\overline{w}}_0}{dt}(t)-{[{\widehat{\overline{\varphi }}}_0(t,\epsilon )]}_1)dt\\ -{\overline{\psi }}_0{(0)}^{\prime }(E_1{\mathsf{\Pi }}_{01}w(0)+E_2{\mathsf{\Pi }}_{00}w(0))+{\overline{\psi }}_0{(T)}^{\prime }{E}_2{Q}_{00}w(0)\\ ={\overline{\psi }}_0{(T)}^{\prime }{E}_2{Q}_{00}w(0)+{\int }_0^T{\overline{\psi }}_0{(t)}^{\prime }(E_2\frac{d{\overline{w}}_0}{dt}(t)-{[{\widehat{\overline{\varphi }}}_0(t,\epsilon )]}_1)dt-{\overline{\psi }}_0{(0)}^{\prime }{E}_2(w^0-{\overline{w}}_0(0)).\end{array}$$

Taking into account this relation and the previous expression for $J_1$, and also dropping terms, which are known after solving the problem ${\overline{P}}_0$, we see that the transformed expression for $J_1$ is the sum ${\mathsf{\Pi }}_{00}J+Q_{00}J$.

Assuming that the problems ${\overline{P}}_0$, ${\mathsf{\Pi }}_{00}P$, $Q_{00}P$ have been solved, we transform by similar way the coefficient $J_2$ in (6). According to (34), $J_2$ has the form:

$$\begin{array}{c}{\int }_0^T{\overline{\mathbb{F}}}_2(t)dt+{\int }_0^{+\infty }{\mathsf{\Pi }}_{01}\mathbb{F}({\tau }_0)d{\tau }_0+{\int }_{-\infty }^0{Q}_{01}\mathbb{F}({\sigma }_0)d{\sigma }_0\\ +{\int }_0^{+\infty }{\mathsf{\Pi }}_{10}\mathbb{F}({\tau }_1)d{\tau }_1+{\int }_{-\infty }^0{Q}_{10}\mathbb{F}({\sigma }_1)d{\sigma }_1.\end{array}$$

Write down the unknown terms in ${\overline{\mathbb{F}}}_2(t)$

$$\begin{array}{c}{\overline{w}}_2{(t)}^{\prime }W(t){\overline{w}}_0(t)+{\overline{u}}_2{(t)}^{\prime }R(t){\overline{u}}_0(t)+{\overline{w}}_1{(t)}^{\prime }(1/2W(t){\overline{w}}_1(t)+{\overline{F}}_{w_0}{(t)}^{\prime })\\ +{\overline{u}}_1{(t)}^{\prime }(1/2R(t){\overline{u}}_1(t)+{\overline{F}}_{u_0}{(t)}^{\prime }).\end{array}$$

Transforming ${\int }_0^T({\overline{w}}_2^{\prime }W{\overline{w}}_0+{\overline{u}}_2^{\prime }R{\overline{u}}_0)dt$ with the help of optimality conditions (18), (19) at $j=0$, integrating by parts, (7) at $j=2$, (20) at $j=0$, and dropping known terms, we obtain $-{\overline{\psi }}_0{(0)}^{\prime }(E_1{\overline{w}}_2(0)+E_2{\overline{w}}_1(0))+{\overline{\psi }}_0{(T)}^{\prime }{E}_2{\overline{w}}_1(T)-{\int }_0^T({\overline{w}}_1^{\prime }{E}_{2}d{\overline{\psi }}_0/dt+{\overline{\psi }}_0^{\prime }({\overline{f}}_{w_0}{\overline{w}}_1+{\overline{f}}_{u_0}{\overline{u}}_1))dt.$

The unknown expression in ${\mathsf{\Pi }}_{01}\mathbb{F}({\tau }_0)$ is

$$\begin{array}{c}{\mathsf{\Pi }}_{01}w{({\tau }_0)}^{\prime }W(0)({\overline{w}}_0(0)+{\mathsf{\Pi }}_{00}w({\tau }_0))+{\mathsf{\Pi }}_{00}w{({\tau }_0)}^{\prime }W(0){\overline{w}}_1(0)\\ +{\mathsf{\Pi }}_{01}u{({\tau }_0)}^{\prime }R(0)({\overline{u}}_0(0)+{\mathsf{\Pi }}_{01}u({\tau }_0))+{\mathsf{\Pi }}_{00}u{({\tau }_0)}^{\prime }R(0){\overline{u}}_1(0).\end{array}$$

The integral of this expression will be transformed using control optimality conditions for problems ${\overline{P}}_0$ and ${\mathsf{\Pi }}_{00}P$, Equations (7) at $j=1$, (8) at $j=1,2$, the formula of integration by parts and Remark 1. Dropping known terms, we have $-{\mathsf{\Pi }}_{00}\psi {(0)}^{\prime }((E_1+E_2){\overline{w}}_1(0)+E_2{\mathsf{\Pi }}_{01}w(0))-{\overline{\psi }}_0{(0)}^{\prime }(E_2{\mathsf{\Pi }}_{01}w(0)+E_1{\mathsf{\Pi }}_{02}w(0))$.

Similarly, we transform the third integral in $J_2$, depending on an unknown expression

$$\begin{array}{c}{\int }_{-\infty }^0(Q_{01}w{({\sigma }_0)}^{\prime }W(T)({\overline{w}}_0(T)+Q_{00}w({\sigma }_0))+Q_{00}w{({\sigma }_0)}^{\prime }W(T){\overline{w}}_1(T)\\ +Q_{01}u{({\sigma }_0)}^{\prime }R(T)({\overline{u}}_0(T)+Q_{00}u({\sigma }_0))+Q_{00}u{({\sigma }_0)}^{\prime }{R}_0(T){\overline{u}}_1(T))d{\sigma }_0\\ =Q_{00}\psi {(0)}^{\prime }((E_1+E_2){\overline{w}}_1(T)+E_2{Q}_{01}w(0))+{\overline{\psi }}_0{(T)}^{\prime }{E}_2{Q}_{01}w(0).\end{array}$$

The unknown expression in ${\mathsf{\Pi }}_{10}\mathbb{F}({\tau }_1)$ is

$$\begin{array}{c}{\mathsf{\Pi }}_{10}w{({\tau }_1)}^{\prime }W(0)({\overline{w}}_0(0)+{\mathsf{\Pi }}_{00}w(0))+{\mathsf{\Pi }}_{10}u{({\tau }_1)}^{\prime }R(0)({\overline{u}}_0(0)+{\mathsf{\Pi }}_{00}u(0))\\ +1/2({\mathsf{\Pi }}_{10}w{({\tau }_1)}^{\prime }W(0){\mathsf{\Pi }}_{10}w({\tau }_1)+{\mathsf{\Pi }}_{10}u{({\tau }_1)}^{\prime }R(0){\mathsf{\Pi }}_{10}u({\tau }_1)).\end{array}$$

Transform the integral

$$\begin{array}{c}{\int }_0^{+\infty }({\mathsf{\Pi }}_{10}w{({\tau }_1)}^{\prime }W(0)({\overline{w}}_0(0)+{\mathsf{\Pi }}_{00}w(0))+{\mathsf{\Pi }}_{10}u{({\tau }_1)}^{\prime }R(0)({\overline{u}}_0(0)+{\mathsf{\Pi }}_{00}u(0)))d{\tau }_1\end{array}$$

with the help of optimality conditions for problems ${\overline{P}}_0$, ${\mathsf{\Pi }}_{00}P$, (10) at $j=0,1,2$, (12) and integration by parts. Dropping known terms, we have $-{\overline{\psi }}_0{(0)}^{\prime }(E_1{\mathsf{\Pi }}_{12}w(0)+E_2{\mathsf{\Pi }}_{11}w(0)+E_3{\mathsf{\Pi }}_{10}w(0))-{\mathsf{\Pi }}_{00}\psi {(0)}^{\prime }(E_2{\mathsf{\Pi }}_{11}w(0)+E_3{\mathsf{\Pi }}_{10}w(0)).$

Transforming in a similar way the fifth integral in $J_2$, depending on unknown terms, we obtain

$$\begin{array}{c}{\int }_{-\infty }^0(Q_{10}w{({\sigma }_1)}^{\prime }W(T)({\overline{w}}_0(T)+Q_{00}w(0))+Q_{10}u{({\sigma }_1)}^{\prime }R(T)({\overline{u}}_0(T)+Q_{00}u(0))\\ +1/2(Q_{10}w{({\sigma }_1)}^{\prime }W(T)Q_{10}w({\sigma }_1)+Q_{10}u{({\sigma }_1)}^{\prime }R(T)Q_{10}u({\sigma }_1)))d{\sigma }_1\\ ={\overline{\psi }}_0{(T)}^{\prime }(E_2{Q}_{11}w(0)+E_3{Q}_{10}w(0))+Q_{00}\psi {(0)}^{\prime }(E_2{Q}_{11}w(0)+E_3{Q}_{10}w(0))\\ +1/2{\int }_{-\infty }^0(Q_{10}w{({\sigma }_1)}^{\prime }W(T)Q_{10}w({\sigma }_1)+Q_{10}u{({\sigma }_1)}^{\prime }R(T)Q_{10}u({\sigma }_1))d{\sigma }_1.\end{array}$$

Substituting the transformed relations into $J_2$, taking into account the second equality in (13), (14) at $j=2$, (16) at $j=1$, (17) and (23) at $j=0$, and also Remark 1, and finally dropping known terms, we obtain the theorem statement for the coefficient $J_2$.

Introduce the notation

$$\begin{array}{c}\vartheta (t,\epsilon )-{\tilde{\vartheta }}_{j-1}(t,\epsilon )={\Delta }_j\overline{\vartheta }(t,\epsilon )+{\displaystyle \sum _{i=0}^1}({\Delta }_j{\mathsf{\Pi }}_i\vartheta ({\tau }_i,\epsilon )+{\Delta }_j{Q}_i\vartheta ({\sigma }_i,\epsilon )),\end{array}$$

(35)

where ${\Delta }_j\overline{\vartheta }(t,\epsilon )=\overline{\vartheta }(t,\epsilon )-{\tilde{\overline{\vartheta }}}_{j-1}(t,\epsilon )={\epsilon }^j{\overline{\vartheta }}_j(t)+\alpha ({\epsilon }^{j+1})$, ${\Delta }_j{\mathsf{\Pi }}_i\vartheta ({\tau }_i,\epsilon )={\mathsf{\Pi }}_i\vartheta ({\tau }_i,\epsilon )-{\tilde{\mathsf{\Pi }}}_{i(j-1)}\vartheta ({\tau }_i,\epsilon )={\epsilon }^j{\mathsf{\Pi }}_{ij}\vartheta ({\tau }_i)+\alpha ({\epsilon }^{j+1})$, ${\Delta }_j{Q}_i\vartheta ({\sigma }_i,\epsilon )=Q_i\vartheta ({\tau }_i,\epsilon )-{\tilde{Q}}_{i(j-1)}\vartheta ({\sigma }_i,\epsilon )={\epsilon }^j{Q}_{ij}\vartheta ({\sigma }_i)+\alpha ({\epsilon }^{j+1})$, $i=0,1$, $\alpha ({\epsilon }^{j+1})$ is a sum of the expansion terms of order ${\epsilon }^{j+1}$ and higher.

Assuming that the problems ${\overline{P}}_j$, ${\mathsf{\Pi }}_{0j}P$, $Q_{0j}P$ and ${\mathsf{\Pi }}_{1(j-1)}P$, $Q_{1(j-1)}P$, $j=\overline{0,n-1}$ have been solved, we will transform each term in the coefficient $J_{2n}$, having the presentation (34) with $k=2n$.

Using the notation (35), we can see that the unknown terms in ${\overline{\mathbb{F}}}_{2n}(t)$ are the following:

$$\begin{array}{c}{\overline{w}}_n{(t)}^{\prime }(1/2W(t){\overline{w}}_n(t)+{[{\widehat{\overline{F}}}_{w(n-1)}{(t,\epsilon )}^{\prime }]}_{n-1})\\ +{\overline{u}}_n{(t)}^{\prime }(1/2R(t){\overline{u}}_n(t)+{[{\widehat{\overline{F}}}_{u(n-1)}{(t,\epsilon )}^{\prime }]}_{n-1})\\ +{[{\Delta }_{n+1}\overline{w}{(t,\epsilon )}^{\prime }(W(t){\tilde{\overline{w}}}_{n-1}(t,\epsilon )+{\{\epsilon {\widehat{\overline{F}}}_{w(n-1)}{(t,\epsilon )}^{\prime }\}}_{n-1})]}_{2n}\\ +{[{\Delta }_{n+1}\overline{u}{(t,\epsilon )}^{\prime }(R(t){\tilde{\overline{u}}}_{n-1}(t,\epsilon )+{\{\epsilon {\widehat{\overline{F}}}_{u(n-1)}{(t,\epsilon )}^{\prime }\}}_{n-1})]}_{2n}.\end{array}$$

Multiplying the Equations (18), (19) by ${\epsilon }^j$, $j=\overline{0,k}$, and summing up the obtained equations, we obtain the following relations

$$\begin{array}{c}{\{R(t){\tilde{\overline{u}}}_k(t,\epsilon )+\epsilon {\widehat{\overline{F}}}_{u(k-1)}{(t,\epsilon )}^{\prime }\}}_k={\{B{(t)}^{\prime }{\tilde{\overline{\psi }}}_k(t,\epsilon )+\epsilon {\widehat{\overline{f}}}_{u(k-1)}{(t,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{k-1}(t,\epsilon )\}}_k,\\ {\{W(t){\tilde{\overline{w}}}_k(t,\epsilon )+\epsilon {\widehat{\overline{F}}}_{w(k-1)}{(t,\epsilon )}^{\prime }\}}_k={\{\mathcal{E}(\epsilon )\frac{d{\tilde{\overline{\psi }}}_k(t,\epsilon )}{dt}\}}_k\\ +{\{A{(t)}^{\prime }{\tilde{\overline{\psi }}}_k(t,\epsilon )+\epsilon {\widehat{\overline{f}}}_{w(k-1)}{(t,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{k-1}(t,\epsilon )\}}_k.\end{array}$$

(36)

Substituting $\vartheta (t,\epsilon )$ from (35) with $j=n+1$ into (2) and equating terms depending on t, we obtain the equation

$$\begin{array}{c}\mathcal{E}(\epsilon )(\frac{d{\tilde{\overline{w}}}_n(t,\epsilon )}{dt}+\frac{d{\Delta }_{n+1}\overline{w}(t,\epsilon )}{dt})=A(t)({\tilde{\overline{w}}}_n(t,\epsilon )+{\Delta }_{n+1}\overline{w}(t,\epsilon ))\\ +B(t)({\tilde{\overline{u}}}_n(t,\epsilon )+{\Delta }_{n+1}\overline{u}(t,\epsilon ))+\epsilon f({\tilde{\overline{\vartheta }}}_n(t,\epsilon )+{\Delta }_{n+1}\overline{\vartheta }(t,\epsilon ),t,\epsilon ).\end{array}$$

(37)

We will use the next easily proved formula from [29], which is valid for any sufficiently smooth vector functions $a(t,\epsilon )$, $b(t,\epsilon )$ and a matrix $\mathbb{D}(t,\epsilon )$ of the corresponding size,

$$\begin{array}{c}{}_k={[{\{b(t,\epsilon )\}}_l^{\prime }\mathbb{D}(t,\epsilon )a(t,\epsilon )]}_k\\ -{[{\{b(t,\epsilon )\}}_l^{\prime }\mathbb{D}(t,\epsilon ){\{a(t,\epsilon )\}}_{k-l-1}]}_k,k,l\in IN,k⩾l.\end{array}$$

(38)

Using (36) with $k=n-1$, (37), (38) with $l=n-1$, $k=2n$, we can rewrite

$$\begin{array}{c}{\int }_0^T({[{\Delta }_{n+1}\overline{w}{(t,\epsilon )}^{\prime }(W(t){\tilde{\overline{w}}}_{n-1}(t,\epsilon )+{\{\epsilon {\widehat{\overline{F}}}_{w(n-1)}{(t,\epsilon )}^{\prime }\}}_{n-1})]}_{2n}\\ +{[{\Delta }_{n+1}\overline{u}{(t,\epsilon )}^{\prime }(R(t){\tilde{\overline{u}}}_{n-1}(t,\epsilon )+{\{\epsilon {\widehat{\overline{F}}}_{u(n-1)}{(t,\epsilon )}^{\prime }\}}_{n-1})]}_{2n})dt\end{array}$$

in the following way

$$\begin{array}{c}{\int }_0^T({[{\Delta }_{n+1}\overline{w}{(t,\epsilon )}^{\prime }{\{\mathcal{E}(\epsilon )\frac{d{\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )}{dt}\}}_{n-1}]}_{2n}+[{\tilde{\overline{\psi }}}_{n-1}{(t,\epsilon )}^{\prime }(A(t){\Delta }_{n+1}\overline{w}(t,\epsilon )\\ +B(t){\Delta }_{n+1}\overline{u}{(t,\epsilon )]}_{2n}+{[{\Delta }_{n+1}\overline{w}{(t,\epsilon )}^{\prime }{\{\epsilon {\widehat{\overline{f}}}_{w(n-1)}{(t,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )\}}_{n-1}]}_{2n}\\ +[{\Delta }_{n+1}\overline{u}{(t,\epsilon )}^{\prime }{\{\epsilon {\widehat{\overline{f}}}_{u(n-1)}{(t,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )\}}_{n-1}\rangle {]}_{2n})dt\\ ={\int }_0^T({[{\Delta }_{n+1}\overline{w}{(t,\epsilon )}^{\prime }{\{\mathcal{E}(\epsilon )\frac{d{\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )}{dt}\}}_{n-1}]}_{2n}+[{\tilde{\overline{\psi }}}_{n-1}{(t,\epsilon )}^{\prime }(\mathcal{E}(\epsilon )(\frac{d{\Delta }_{n+1}\overline{w}(t,\epsilon )}{dt}\\ +\frac{d{\tilde{\overline{w}}}_n(t,\epsilon )}{dt})-A(t){\tilde{\overline{w}}}_n(t,\epsilon )-B(t){\tilde{\overline{u}}}_n(t,\epsilon )-\epsilon f({\tilde{\overline{\vartheta }}}_n(t,\epsilon )+{\Delta }_{n+1}\overline{\vartheta }(t,\epsilon ),t,\epsilon ){)]}_{2n}\\ +{[{\tilde{\overline{\psi }}}_{n-1}{(t,\epsilon )}^{\prime }({\{\epsilon {\widehat{\overline{f}}}_{w(n-1)}(t,\epsilon )\}}_{n-1}{\Delta }_{n+1}\overline{w}(t,\epsilon )+{\{\epsilon {\widehat{\overline{f}}}_{u(n-1)}(t,\epsilon )\}}_{n-1}{\Delta }_{n+1}\overline{u}(t,\epsilon ))]}_{2n})dt.\end{array}$$

Integrating by parts in the first term of the last expression, taking into account the equality ${\Delta }_{n+1}\overline{\vartheta }(t,\epsilon )={\Delta }_n\overline{\vartheta }(t,\epsilon )-{\epsilon }^n{\overline{\vartheta }}_n(t)$, decomposing $f({\tilde{\overline{\vartheta }}}_n(t,\epsilon )+{\Delta }_{n+1}\overline{\vartheta }(t,\epsilon ),t,\epsilon )$ in the neighborhood of ${\tilde{\overline{\vartheta }}}_{n-1}(t,\epsilon )$, and omitting known terms, we obtain

$$\begin{array}{c}{[{\Delta }_{n+1}\overline{w}{(t,\epsilon )}^{\prime }\mathcal{E}(\epsilon ){\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )]}_{2n}{{|}_0^T+({\overline{\psi }}_{n-1}{(t)}^{\prime }{E}_2{\overline{w}}_n(t)+{\overline{\psi }}_{n-2}{(t)}^{\prime }{E}_3{\overline{w}}_n(t))|}_0^T\\ +{\int }_0^T({\overline{w}}_n^{\prime }(-E_2\frac{d{\overline{\psi }}_{n-1}}{dt}-E_3\frac{d{\overline{\psi }}_{n-2}}{dt}-{[{\{{\tilde{\overline{\psi }}}_{n-1}{(t,\epsilon )}^{\prime }\epsilon {\widehat{\overline{f}}}_{\vartheta (n-1)}(t,\epsilon )\}}_n({\epsilon }^n{\overline{\vartheta }}_n+{\Delta }_{n+1}\vartheta (t,\epsilon ))]}_{2n}\\ +{[{\tilde{\overline{\psi }}}_{n-1}{(t,\epsilon )}^{\prime }({\{\epsilon {\widehat{\overline{f}}}_{w(n-1)}(t,\epsilon )\}}_{n-1}{\Delta }_{n+1}\overline{w}(t,\epsilon )+{\{\epsilon {\widehat{\overline{f}}}_{u(n-1)}(t,\epsilon )\}}_{n-1}{\Delta }_{n+1}\overline{u}(t,\epsilon ))]}_{2n})dt\\ ={[{\Delta }_n\overline{w}{(t,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3){\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )]}_{2n-1}{|}_0^T\\ -{\int }_0^T({\overline{w}}_n{(t)}^{\prime }(E_2\frac{d{\overline{\psi }}_{n-1}}{dt}+E_3\frac{d{\overline{\psi }}_{n-2}}{dt}+{[\epsilon {\widehat{\overline{f}}}_{w(n-1)}{(t,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )]}_n)\\ +[{\overline{u}}_n{(t)}^{\prime }({[\epsilon {\widehat{\overline{f}}}_{u(n-1)}{(t,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )]}_n)dt.\end{array}$$

Taking into account the last relation, omitting known terms, we obtain the following expression for the first term of $J_{2n}$:

$$\begin{array}{c}{\int }_0^T{\overline{\mathbb{F}}}_{2n}(t)dt={[{\Delta }_n\overline{w}{(t,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3){\tilde{\overline{\psi }}}_{n-1}(t,\epsilon )]}_{2n-1}{|}_0^T+{\overline{J}}_n\end{array}$$

$$\begin{array}{c}-{\overline{w}}_n{(T)}^{\prime }{E}_1(Q_{0(n-1)}\psi (0)+Q_{1(n-2)}\psi (0)).\end{array}$$

The next step is the transformation of the unknown parts of ${\mathsf{\Pi }}_{0(2n-1)}\mathbb{F}({\tau }_0)$, which, after substituting (35) and some transformations, is given below

$$\begin{array}{c}{[{\Delta }_n{\mathsf{\Pi }}_{0}w{({\tau }_0,\epsilon )}^{\prime }({\{W(\epsilon {\tau }_0){\tilde{\overline{w}}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}+{\{\epsilon {\widehat{\overline{F}}}_{w(n-1)}{(\epsilon {\tau }_0,\epsilon )}^{\prime }\}}_{n-1})]}_{2n-1}\\ +{[{\Delta }_n{\mathsf{\Pi }}_{0}u{({\tau }_0,\epsilon )}^{\prime }({\{R(\epsilon {\tau }_0){\tilde{\overline{u}}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}+{\{\epsilon {\widehat{\overline{F}}}_{u(n-1)}{(\epsilon {\tau }_0,\epsilon )}^{\prime }\}}_{n-1})]}_{2n-1}\\ +[{({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))}^{\prime }({\{W(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0(n-1)}w({\tau }_0,\epsilon )\}}_{n-1}\\ +{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{F}_w{({\tau }_0,\epsilon )}^{\prime }\}}_{n-1}{)]}_{2n-1}+[({\Delta }_n\overline{u}(\epsilon {\tau }_0,\epsilon )\\ +{\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon ){{)}^{\prime }({\{R(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0(n-1)}u({\tau }_0,\epsilon )\}}_{n-1}+{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{F}_u{({\tau }_0,\epsilon )}^{\prime }\}}_{n-1})]}_{2n-1}.\end{array}$$

Substituting $\vartheta (t,\epsilon )$ from (35) into (2) and considering terms depending on ${\tau }_0$, we obtain the equation

$$\begin{array}{c}(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3)(\frac{d{\tilde{\mathsf{\Pi }}}_{0(n-1)}w({\tau }_0,\epsilon )}{d{\tau }_0}+\frac{d{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon )}{d{\tau }_0})\\ =A(\epsilon {\tau }_0)({\tilde{\mathsf{\Pi }}}_{0(n-1)}w({\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))+B(\epsilon {\tau }_0)({\tilde{\mathsf{\Pi }}}_{0(n-1)}u({\tau }_0,\epsilon )\\ +{\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon ))+\epsilon (f({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta ({\tau }_0,\epsilon )\\ +{\Delta }_n\overline{\vartheta }(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon )-f({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n\overline{\vartheta }(\epsilon {\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon )).\end{array}$$

(39)

Using (36) with $k=n-1$, (39) and (38), we transform the following expression:

$$\begin{array}{c}{\int }_0^{+\infty }({[{\Delta }_n{\mathsf{\Pi }}_{0}w{({\tau }_0,\epsilon )}^{\prime }({\{W(\epsilon {\tau }_0){\tilde{\overline{w}}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}+{\{\epsilon {\widehat{\overline{F}}}_{w(n-1)}{(\epsilon {\tau }_0,\epsilon )}^{\prime }\}}_{n-1})]}_{2n-1}\\ +{[{\Delta }_n{\mathsf{\Pi }}_{0}u{({\tau }_0,\epsilon )}^{\prime }({\{R(\epsilon {\tau }_0){\tilde{\overline{u}}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}+{\{\epsilon {\widehat{\overline{F}}}_{u(n-1)}{(\epsilon {\tau }_0,\epsilon )}^{\prime }\}}_{n-1})]}_{2n-1})d{\tau }_0.\end{array}$$

Omitting known terms, we have

$$\begin{array}{c}{\int }_0^{+\infty }([{\Delta }_n{\mathsf{\Pi }}_{0}w{({\tau }_0,\epsilon )}^{\prime }({\{\mathcal{E}(\epsilon )\frac{d{\tilde{\overline{\psi }}}_{n-1}}{dt}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}+{\{A{(\epsilon {\tau }_0)}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}\\ +{\{\epsilon {\widehat{\overline{f}}}_{w(n-1)}{(\epsilon {\tau }_0,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}{)]}_{2n-1}+[{\Delta }_n{\mathsf{\Pi }}_{0}u{({\tau }_0,\epsilon )}^{\prime }({\{B{(\epsilon {\tau }_0)}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}\\ +{\{\epsilon {\widehat{\overline{f}}}_{u(n-1)}{(\epsilon {\tau }_0,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-1}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}{)]}_{2n-1})d{\tau }_0={\int }_0^{+\infty }({\mathsf{\Pi }}_{0n}{w}^{\prime }{E}_1\frac{d{\overline{\psi }}_{n-1}}{dt}(0)\\ +{[{\Delta }_n{\mathsf{\Pi }}_{0}w{({\tau }_0,\epsilon )}^{\prime }{\{(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3)\frac{d{\tilde{\overline{\psi }}}_{n-2}}{dt}(\epsilon {\tau }_0,\epsilon )\}}_{n-2}]}_{2n-2}\\ +{[{\tilde{\overline{\psi }}}_{n-1}{(\epsilon {\tau }_0,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3)\frac{d{\Delta }_n{\mathsf{\Pi }}_{0}w}{d{\tau }_0}({\tau }_0,\epsilon )]}_{2n-1}\\ -[{\tilde{\overline{\psi }}}_{n-1}{(\epsilon {\tau }_0,\epsilon )}^{\prime }(\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_{\vartheta }({\tau }_0,\epsilon ){\Delta }_n\overline{\vartheta }(\epsilon {\tau }_0,\epsilon )+\epsilon f_{\vartheta }({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon ){\Delta }_n{\mathsf{\Pi }}_0\vartheta ({\tau }_0,\epsilon ){)]}_{2n-1}+[{\tilde{\overline{\psi }}}_{n-1}{(\epsilon {\tau }_0,\epsilon )}^{\prime }(\epsilon {\widehat{\overline{f}}}_{w(n-1)}(\epsilon {\tau }_0,\epsilon ){\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon )\\ +\epsilon {\widehat{\overline{f}}}_{u(n-1)}(\epsilon {\tau }_0,\epsilon ){\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon ){)]}_{2n-1})d{\tau }_0.\end{array}$$

From here, applying the formula of integrating by parts and Remark 1, omitting known terms, we obtain

$$\begin{array}{c}-{[{\Delta }_n{\mathsf{\Pi }}_{0}w{(0,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3){\tilde{\overline{\psi }}}_{n-1}(0,\epsilon )]}_{2n-1}\\ -{\int }_0^{+\infty }([{\tilde{\overline{\psi }}}_{n-1}{(\epsilon {\tau }_0,\epsilon )}^{\prime }{(\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_{\vartheta }({\Delta }_n\overline{\vartheta }(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta ({\tau }_0,\epsilon ))]}_{2n-1})d{\tau }_0.\end{array}$$

Multiplying the Equations (21), (22) by ${\epsilon }^j$, $j=\overline{0,k}$, and summing up the obtained equations, we obtain the equalities

$$\begin{array}{c}{\{R(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0k}u\}}_k+{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(k-1)}{F}_u{({\tau }_0,\epsilon )}^{\prime }\}}_k\\ ={\{B{(\epsilon {\tau }_0)}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0k}\psi \}}_k+{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(k-1)}{f}_u{({\tau }_0,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{k-1}(\epsilon {\tau }_0,\epsilon )\}}_k\\ +{\{\epsilon f_u{({\tilde{\overline{\vartheta }}}_{k-1}(\epsilon {\tau }_0,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(k-1)}\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(k-1)}\psi \}}_k,\\ {\{W(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0k}w\}}_k+{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(k-1)}{F}_w{({\tau }_0,\epsilon )}^{\prime }\}}_k={\{(E_1+E_2+\epsilon E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{0k}\psi }{d{\tau }_0}\}}_k\\ +{\{A{(\epsilon {\tau }_0)}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0k}\psi \}}_k+{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(k-1)}{f}_w{({\tau }_0,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{k-1}(\epsilon {\tau }_0,\epsilon )\}}_k\\ +{\{\epsilon f_w{({\tilde{\overline{\vartheta }}}_{k-1}(\epsilon {\tau }_0,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(k-1)}\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(k-1)}\psi \}}_k.\end{array}$$

(40)

We transform

$$\begin{array}{c}{\int }_0^{+\infty }([{({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))}^{\prime }({\{W(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0(n-1)}w({\tau }_0,\epsilon )\}}_{n-1}\\ +{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{F}_w{({\tau }_0,\epsilon )}^{\prime }\}}_{n-1}{)]}_{2n-1}+[({\Delta }_n\overline{u}(\epsilon {\tau }_0,\epsilon )\\ +{\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon ){)}^{\prime }({\{R(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0(n-1)}u({\tau }_0,\epsilon )\}}_{n-1}+{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{F}_u{({\tau }_0,\epsilon )}^{\prime }\}}_{n-1}){]}_{2n-1})d{\tau }_0.\end{array}$$

Using (40) and (38), as a result, we obtain

$$\begin{array}{c}{\int }_0^{+\infty }({[{({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))}^{\prime }{\{(E_1+E_2+\epsilon E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{0(n-1)}\psi }{d{\tau }_0}\}}_{n-1}]}_{2n-1}\\ +{[{\tilde{\mathsf{\Pi }}}_{0(n-1)}{\psi }^{\prime }(\epsilon E_1+E_2+E_3)(A(\epsilon {\tau }_0){\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+B(\epsilon {\tau }_0){\Delta }_n\overline{u}(\epsilon {\tau }_0,\epsilon ))]}_{2n-1}\\ +{[{\tilde{\mathsf{\Pi }}}_{0(n-1)}{\psi }^{\prime }(\epsilon E_1+E_2+E_3)(A(\epsilon {\tau }_0){\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon )+B(\epsilon {\tau }_0){\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon ))]}_{2n-1}\\ +[{\tilde{\overline{\psi }}}_{n-1}{(\epsilon {\tau }_0,\epsilon )}^{\prime }({\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_w({\tau }_0,\epsilon )\}}_{n-1}({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))\\ +{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_u({\tau }_0,\epsilon )\}}_{n-1}({\Delta }_n\overline{u}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon )){)]}_{2n-1}\\ +[{\tilde{\mathsf{\Pi }}}_{0(n-1)}{\psi }^{\prime }(\epsilon E_1+E_2+E_3)({\{\epsilon f_w({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta ,\epsilon {\tau }_0,\epsilon )\}}_{n-1}({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )\\ +{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))+{\{\epsilon f_u({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta ,\epsilon {\tau }_0,\epsilon )\}}_{n-1}({\Delta }_n\overline{u}(\epsilon {\tau }_0,\epsilon )\\ +{\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon ){))]}_{2n-1})d{\tau }_0.\end{array}$$

In view of (37) and (39), we obtain from the last expression, omitting known terms, the following:

$$\begin{array}{c}{\int }_0^{+\infty }({[{({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))}^{\prime }{\{(E_1+E_2+\epsilon E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{0(n-1)}\psi ({\tau }_0,\epsilon )}{d{\tau }_0}\}}_{n-1}]}_{2n-1}\\ +[{\tilde{\mathsf{\Pi }}}_{0(n-1)}\psi {({\tau }_0,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3)((E_1+\epsilon E_2+{\epsilon }^2{E}_3)(\frac{d{\tilde{\overline{w}}}_{n-1}(\epsilon {\tau }_0,\epsilon )}{dt}\\ +\frac{d{\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )}{dt})-A(\epsilon {\tau }_0){\tilde{\overline{w}}}_{n-1}(\epsilon {\tau }_0,\epsilon )-B(\epsilon {\tau }_0){\tilde{\overline{u}}}_{n-1}(\epsilon {\tau }_0,\epsilon )\\ -{\{\epsilon {\widehat{\overline{f}}}_{\vartheta (n-1)}(\epsilon {\tau }_0,\epsilon )\}}_{n-1}{\Delta }_n\overline{\vartheta }(\epsilon {\tau }_0,\epsilon ){)]}_{2n-1}\\ +[{\tilde{\mathsf{\Pi }}}_{0(n-1)}\psi {({\tau }_0,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3)((\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3)(\frac{d{\tilde{\mathsf{\Pi }}}_{0(n-1)}w({\tau }_0,\epsilon )}{d{\tau }_0}+\frac{d{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon )}{d{\tau }_0})\\ -A(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0(n-1)}w({\tau }_0,\epsilon )-B(\epsilon {\tau }_0){\tilde{\mathsf{\Pi }}}_{0(n-1)}u({\tau }_0,\epsilon )-{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_{\vartheta }({\tau }_0,\epsilon )\}}_{n-1}{\Delta }_n\overline{\vartheta }(\epsilon {\tau }_0,\epsilon )\\ -{\{\epsilon f_{\vartheta }({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0,\epsilon )\}}_{n-1}{\Delta }_n{\mathsf{\Pi }}_0\vartheta ({\tau }_0,\epsilon ){)]}_{2n-1}\\ +[{\tilde{\overline{\psi }}}_{n-1}{(\epsilon {\tau }_0,\epsilon )}^{\prime }({\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_w({\tau }_0,\epsilon )\}}_{n-1}({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))\end{array}$$

$$\begin{array}{c}+{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_u({\tau }_0,\epsilon )\}}_{n-1}({\Delta }_n\overline{u}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon )){)]}_{2n-1}\\ +[{\tilde{\mathsf{\Pi }}}_{0(n-1)}\psi {({\tau }_0,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3)(\{\epsilon f_w({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0{,\epsilon )\}}_{n-1}({\Delta }_n\overline{w}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w({\tau }_0,\epsilon ))+\{\epsilon f_u({\tilde{\overline{\vartheta }}}_{n-1}(\epsilon {\tau }_0,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta ({\tau }_0,\epsilon ),\epsilon {\tau }_0{,\epsilon )\}}_{n-1}({\Delta }_n\overline{u}(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}u({\tau }_0,\epsilon )){)]}_{2n-1})d{\tau }_0.\end{array}$$

Integrating by parts in the last expression and dropping known terms, we obtain

$$\begin{array}{c}-{[{({\Delta }_n\overline{w}(0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w(0,\epsilon ))}^{\prime }(E_1+E_2+\epsilon E_3){\tilde{\mathsf{\Pi }}}_{0(n-1)}\psi (0,\epsilon )]}_{2n-1}\\ +{\int }_0^{+\infty }{[{\tilde{\overline{\psi }}}_{n-1}{(\epsilon {\tau }_0,\epsilon )}^{\prime }{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-1)}{f}_{\vartheta }({\tau }_0,\epsilon )\}}_{n-1}({\Delta }_n\overline{\vartheta }(\epsilon {\tau }_0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta ({\tau }_0,\epsilon ))]}_{2n-1}d{\tau }_0.\end{array}$$

Summing up the results, obtained from the transformed terms of the integral

${\int }_0^{+\infty }{\mathsf{\Pi }}_{0(2n-1)}\mathbb{F}d{\tau }_0$, after dropping known terms, we have

$$\begin{array}{c}-{[{\Delta }_n{\mathsf{\Pi }}_{0}w{(0,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3){\tilde{\overline{\psi }}}_{n-1}(0,\epsilon )]}_{2n-1}\\ -{[{({\Delta }_n\overline{w}(0,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{0}w(0,\epsilon ))}^{\prime }(E_1+E_2+\epsilon E_3){\tilde{\mathsf{\Pi }}}_{0(n-1)}\psi (0,\epsilon )]}_{2n-1}.\end{array}$$

Performing similar transformations for ${\int }_{-\infty }^0{Q}_{0(2n-1)}\mathbb{F}d{\sigma }_0$, we obtain the following result:

$$\begin{array}{c}{[{\Delta }_n{Q}_{0}w{(0,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+E_2+\epsilon E_3){\tilde{\overline{\psi }}}_{n-1}(T,\epsilon )]}_{2n-1}\\ +{[{({\Delta }_n\overline{w}(T,\epsilon )+{\Delta }_n{Q}_{0}w(0,\epsilon ))}^{\prime }(E_1+E_2+\epsilon E_3){\tilde{Q}}_{0(n-1)}\psi (0,\epsilon )]}_{2n-1}.\end{array}$$

Furthermore, we apply the analogous transformations for the forth term of $J_{2n}$. The integral over the interval $[0,+\infty )$ of unknown terms of ${\mathsf{\Pi }}_{1(2n-2)}\mathbb{F}({\tau }_1)$ is presented as the sum

$$\begin{array}{c}{\displaystyle \sum _{i=1}^4}{s}_i+{\int }_0^{+\infty }({\mathsf{\Pi }}_{1(n-1)}w{({\tau }_1)}^{\prime }({[W({\epsilon }^2{\tau }_1){\tilde{\overline{w}}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}+{[{\widehat{\overline{F}}}_{w(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }]}_{n-2})\\ +{\mathsf{\Pi }}_{1(n-1)}u{({\tau }_1)}^{\prime }({[R({\epsilon }^2{\tau }_1){\tilde{\overline{u}}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}+{[{\widehat{\overline{F}}}_{u(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }]}_{n-2}))d{\tau }_1\\ +{\int }_0^{+\infty }({\mathsf{\Pi }}_{1(n-1)}w{({\tau }_1)}^{\prime }({[W({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{0(n-1)}w(\epsilon {\tau }_1,\epsilon )]}_{n-1}+{[{\widehat{\mathsf{\Pi }}}_{0(n-2)}{F}_w{(\epsilon {\tau }_1,\epsilon )}^{\prime }]}_{n-2})\\ +{\mathsf{\Pi }}_{1(n-1)}u{({\tau }_1)}^{\prime }({[R({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{0(n-1)}u(\epsilon {\tau }_1,\epsilon )]}_{n-1}+{[{\widehat{\mathsf{\Pi }}}_{0(n-2)}{F}_u{(\epsilon {\tau }_1,\epsilon )}^{\prime }]}_{n-2}))d{\tau }_1\\ +{\int }_0^{+\infty }({\mathsf{\Pi }}_{1(n-1)}w{({\tau }_1)}^{\prime }(1/2W(0){\mathsf{\Pi }}_{1(n-1)}w({\tau }_1)+{[W({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{1(n-2)}w({\tau }_1,\epsilon )]}_{n-1}\\ +{[{\widehat{\mathsf{\Pi }}}_{1(n-2)}{F}_w{({\tau }_1,\epsilon )}^{\prime }]}_{n-2})+{\mathsf{\Pi }}_{1(n-1)}u{({\tau }_1)}^{\prime }(1/2R(0){\mathsf{\Pi }}_{1(n-1)}u({\tau }_1)\\ +{[R({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{1(n-2)}u({\tau }_1,\epsilon )]}_{n-1}+{[{\widehat{\mathsf{\Pi }}}_{1(n-2)}{F}_u{({\tau }_1,\epsilon )}^{\prime }]}_{n-2}))d{\tau }_1,\end{array}$$

(41)

where the expressions for $s_i$, $i=\overline{1,4}$, will be written later when they will be transformed.

Substituting $\vartheta (t,\epsilon )$ from (35) into (2) and considering terms depending on ${\tau }_1$, we obtain the equation

$$\begin{array}{c}(\frac{1}{{\epsilon }^2}{E}_1+\frac{1}{\epsilon }{E}_2+E_3)(\frac{d{\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )}{d{\tau }_1}+\frac{d{\Delta }_n{\mathsf{\Pi }}_{1}w({\tau }_1,\epsilon )}{d{\tau }_1})\\ =A({\epsilon }^2{\tau }_1)({\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_{1}w({\tau }_1,\epsilon ))+B({\epsilon }^2{\tau }_1)({\tilde{\mathsf{\Pi }}}_{1(n-1)}u({\tau }_1,\epsilon )\\ +{\Delta }_n{\mathsf{\Pi }}_{1}u({\tau }_1,\epsilon ))+\epsilon (f({\tilde{\overline{\vartheta }}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta (\epsilon {\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{1(n-1)}\vartheta ({\tau }_1,\epsilon )\\ +{\Delta }_n\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )\\ -f({\tilde{\overline{\vartheta }}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta (\epsilon {\tau }_1,\epsilon )+{\Delta }_n\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )).\end{array}$$

(42)

Using (36) with $k=n-2$ and $t={\epsilon }^2{\tau }_1$ in the expression

$$\begin{array}{c}{s}_1={\int }_0^{+\infty }({[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }({\{W({\epsilon }^2{\tau }_1){\tilde{\overline{w}}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}+{\{\epsilon {\widehat{\overline{F}}}_{w(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }\}}_{n-2})]}_{2n-2}\\ +{[{\Delta }_n{\mathsf{\Pi }}_{1}u{({\tau }_1,\epsilon )}^{\prime }({\{R({\epsilon }^2{\tau }_1){\tilde{\overline{u}}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}+{\{\epsilon {\widehat{\overline{F}}}_{u(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }\}}_{n-2})]}_{2n-2})d{\tau }_1,\end{array}$$

we obtain

$$\begin{array}{c}{\int }_0^{+\infty }{([{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }(E_1+\epsilon E_2+{\epsilon }^2{E}_3)\frac{d{\tilde{\overline{\psi }}}_{n-2}}{dt}({\epsilon }^2{\tau }_1,\epsilon ))]}_{2n-2}\\ +{[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }({\{A{({\epsilon }^2{\tau }_1)}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}+{\{\epsilon {\widehat{\overline{f}}}_{w(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2})]}_{2n-2}\\ +{[{\Delta }_n{\mathsf{\Pi }}_{1}u{({\tau }_1,\epsilon )}^{\prime }({\{B{({\epsilon }^2{\tau }_1)}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}+{\{\epsilon {\widehat{\overline{f}}}_{u(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2})]}_{2n-2})d{\tau }_1.\end{array}$$

Then, applying (38) with $k=2n-2$, $l=n-2$, and (42), we have

$$\begin{array}{c}{\int }_0^{+\infty }({[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }(E_1+\epsilon E_2+{\epsilon }^2{E}_3)\frac{d{\tilde{\overline{\psi }}}_{n-2}}{dt}({\epsilon }^2{\tau }_1,\epsilon )]}_{2n-2}\\ +{[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(A({\epsilon }^2{\tau }_1){\Delta }_n{\mathsf{\Pi }}_{1}w({\tau }_1,\epsilon )+B({\epsilon }^2{\tau }_1){\Delta }_n{\mathsf{\Pi }}_{1}u({\tau }_1,\epsilon ))]}_{2n-2}\\ +{[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(\epsilon {\widehat{\overline{f}}}_{w(n-2)}({\epsilon }^2{\tau }_1,\epsilon ){\Delta }_n{\mathsf{\Pi }}_{1}w({\tau }_1,\epsilon )+\epsilon {\widehat{\overline{f}}}_{u(n-2)}({\epsilon }^2{\tau }_1,\epsilon ){\Delta }_n{\mathsf{\Pi }}_{1}u({\tau }_1,\epsilon ))]}_{2n-2})d{\tau }_1\\ ={\int }_0^{+\infty }({[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }(E_1+\epsilon E_2+{\epsilon }^2{E}_3)\frac{d{\tilde{\overline{\psi }}}_{n-2}}{dt}({\epsilon }^2{\tau }_1,\epsilon )]}_{2n-2}\\ +[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }((\frac{1}{{\epsilon }^2}{E}_1+\frac{1}{\epsilon }{E}_2+E_3)(\frac{d{\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )}{d{\tau }_1}+\frac{d{\Delta }_n{\mathsf{\Pi }}_{1}w({\tau }_1,\epsilon )}{d{\tau }_1})\\ -A({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )-B({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{1(n-1)}u({\tau }_1,\epsilon )-\epsilon (f({\tilde{\overline{\vartheta }}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta (\epsilon {\tau }_1,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{1(n-1)}\vartheta ({\tau }_1,\epsilon )+{\Delta }_n\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )\\ -f({\tilde{\overline{\vartheta }}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-1)}\vartheta (\epsilon {\tau }_1,\epsilon )+{\Delta }_n\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon ){))]}_{2n-2}\\ +{[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }\epsilon {\widehat{\overline{f}}}_{\vartheta (n-2)}({\epsilon }^2{\tau }_1,\epsilon ){\Delta }_n{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon )]}_{2n-2})d{\tau }_1.\end{array}$$

Integrating by parts in the last relation, using Remark 1, dropping known terms, we obtain

$$\begin{array}{c}-{[{\Delta }_n{\mathsf{\Pi }}_{1}w{(0,\epsilon )}^{\prime }(\frac{1}{{\epsilon }^2}{E}_1+\frac{1}{\epsilon }{E}_2+E_3){\tilde{\overline{\psi }}}_{n-2}(0,\epsilon )]}_{2n-2}\\ +{\int }_0^{+\infty }(\underset{\{1\}}{\underbrace{{[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(\frac{1}{{\epsilon }^2}{E}_1+\frac{1}{\epsilon }{E}_2+E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )}{d{\tau }_1}]}_{2n-2}}}\\ +{[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }\epsilon {\widehat{\overline{f}}}_{\vartheta (n-2)}({\epsilon }^2{\tau }_1,\epsilon ){\Delta }_n{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon )]}_{2n-2}\\ -[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(\epsilon {\widehat{\mathsf{\Pi }}}_{1(n-2)}{f}_{\vartheta }({\tau }_1,\epsilon )({\Delta }_n\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon ))\\ +\epsilon (\underset{\{1\}}{\underbrace{{\widehat{\overline{f}}}_{\vartheta (n-2)}({\epsilon }^2{\tau }_1,\epsilon )}}+\underset{\{2\}}{\underbrace{{\widehat{\mathsf{\Pi }}}_{0(n-2)}{f}_{\vartheta }(\epsilon {\tau }_1,\epsilon )}}+{\widehat{\mathsf{\Pi }}}_{1(n-2)}{f}_{\vartheta }({\tau }_1,\epsilon )){\epsilon }^{n-1}{\mathsf{\Pi }}_{1(n-1)}\vartheta ({\tau }_1)\\ +\epsilon f_{\vartheta }({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{1(n-2)}\vartheta ,{\epsilon }^2{\tau }_1,\epsilon ){\Delta }_n{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon ){)]}_{2n-2}\\ -\underset{\{1\}}{\underbrace{{\mathsf{\Pi }}_{1(n-1)}w{({\tau }_1)}^{\prime }{[A{({\epsilon }^2{\tau }_1)}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}}}\\ -\underset{\{1\}}{\underbrace{{\mathsf{\Pi }}_{1(n-1)}u{({\tau }_1)}^{\prime }{[B{({\epsilon }^2{\tau }_1)}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}}})d{\tau }_1.\end{array}$$

(43)

Consider together the first integral in (41) and some terms with ${\mathsf{\Pi }}_{1(n-1)}\vartheta ({\tau }_1,\epsilon )$ in the transformed last expression for $s_1$, marked by $\{1\}$, namely, the expression of the form

$$\begin{array}{c}{\int }_0^{+\infty }({\mathsf{\Pi }}_{1(n-1)}w{({\tau }_1)}^{\prime }({[W({\epsilon }^2{\tau }_1){\tilde{\overline{w}}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}+{[{\widehat{\overline{F}}}_{w(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }]}_{n-2}\\ -{[A{({\epsilon }^2{\tau }_1)}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}-{[{\widehat{\overline{f}}}_{w(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-2})\\ +{\mathsf{\Pi }}_{1(n-1)}u{({\tau }_1)}^{\prime }({[R({\epsilon }^2{\tau }_1){\tilde{\overline{u}}}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}+{[{\widehat{\overline{F}}}_{u(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }]}_{n-2}\\ -{[B{({\epsilon }^2{\tau }_1)}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}-{[{\widehat{\overline{f}}}_{u(n-2)}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-2})\\ +{[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(\frac{1}{{\epsilon }^2}{E}_1+\frac{1}{\epsilon }{E}_2+E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )}{d{\tau }_1}]}_{2n-2})d{\tau }_1.\end{array}$$

Transforming this expression with the help of (36) with $k=n-1$ and (10) at $j=n-1,n,n+1$ and omitting some known terms, we have

$$\begin{array}{c}{\int }_0^{+\infty }({\mathsf{\Pi }}_{1(n-1)}w{({\tau }_1)}^{\prime }{[E_1\frac{d{\tilde{\overline{\psi }}}_{n-1}}{dt}({\epsilon }^2{\tau }_1,\epsilon )+E_2\frac{d{\tilde{\overline{\psi }}}_{n-2}}{dt}({\epsilon }^2{\tau }_1,\epsilon )+E_3\frac{d{\tilde{\overline{\psi }}}_{n-3}}{dt}({\epsilon }^2{\tau }_1,\epsilon )]}_{n-1}\\ +{\overline{\psi }}_{n-1}{(0)}^{\prime }(E_1\frac{d{\mathsf{\Pi }}_{1(n+1)}w({\tau }_1)}{d{\tau }_1}+E_2\frac{d{\mathsf{\Pi }}_{1n}w({\tau }_1)}{d{\tau }_1}+E_3\frac{d{\mathsf{\Pi }}_{1(n-1)}w({\tau }_1)}{d{\tau }_1}\\ +[{\tilde{\overline{\psi }}}_{n-2}{({\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(\frac{1}{{\epsilon }^2}{E}_1+\frac{1}{\epsilon }{E}_2+E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )}{d{\tau }_1}){]}_{2n-2})d{\tau }_1.\end{array}$$

From here, using Remark 1, integrating by parts and omitting known terms, we obtain

$$-{\overline{\psi }}_{n-1}{(0)}^{\prime }(E_1{\mathsf{\Pi }}_{1(n+1)}w(0)+E_2{\mathsf{\Pi }}_{1n}w(0)+E_3{\mathsf{\Pi }}_{1(n-1)}w(0)).$$

Further changes concern the expression

$$\begin{array}{c}{s}_2={\int }_0^{+\infty }([{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }({\{W({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{0(n-2)}w(\epsilon {\tau }_1,\epsilon )\}}_{n-2}\\ +{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-2)}{F}_w(\epsilon {\tau }_1,\epsilon )\}}_{n-2}{)]}_{2n-2}+[{\Delta }_n{\mathsf{\Pi }}_{1}u{({\tau }_1,\epsilon )}^{\prime }({\{R({\epsilon }^2{\tau }_1,\epsilon ){\tilde{\mathsf{\Pi }}}_{0(n-2)}u(\epsilon {\tau }_1,\epsilon )\}}_{n-2}\\ +{\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-2)}{F}_u(\epsilon {\tau }_1,\epsilon )\}}_{n-2}{)]}_{2n-2})d{\tau }_1.\end{array}$$

It will be transformed using (40) with $k=n-2$ and (38) in the following way:

$$\begin{array}{c}{\int }_0^{+\infty }({[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }(E_1+E_2+\epsilon E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon )}{d{\tau }_0}]}_{2n-2}\\ +{[{\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi {(\epsilon {\tau }_1,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3)(A({\epsilon }^2{\tau }_1){\Delta }_n{\mathsf{\Pi }}_{1}w({\tau }_1,\epsilon )+B({\epsilon }^2{\tau }_1){\Delta }_n{\mathsf{\Pi }}_{1}u({\tau }_1,\epsilon ))]}_{2n-2}\\ +[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }({\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-2)}{f}_w{(\epsilon {\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}\\ +{\{\epsilon f_w{({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon )\}}_{n-2}{)]}_{2n-2}\\ +[{\Delta }_n{\mathsf{\Pi }}_{1}u{({\tau }_1,\epsilon )}^{\prime }({\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-2)}{f}_u{(\epsilon {\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}+\{\epsilon f_u({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1{,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon ){\}}_{n-2}{)]}_{2n-2})d{\tau }_1.\end{array}$$

From here, using (42) and omitting some known terms, we have

$$\begin{array}{c}{\int }_0^{+\infty }({[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }(E_1+E_2+\epsilon E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon )}{d{\tau }_0}]}_{2n-2}+[{\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi {(\epsilon {\tau }_1,\epsilon )}^{\prime }(\epsilon E_1\\ +E_2+E_3)((\frac{1}{{\epsilon }^2}{E}_1+\frac{1}{\epsilon }{E}_2+E_3)(\frac{d{\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )}{d{\tau }_1}+\frac{d{\Delta }_n{\mathsf{\Pi }}_{1}w({\tau }_1,\epsilon )}{d{\tau }_1})\end{array}$$

$$\begin{array}{c}-A({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )-B({\epsilon }^2{\tau }_1){\tilde{\mathsf{\Pi }}}_{1(n-1)}u({\tau }_1,\epsilon )-\epsilon {\widehat{\mathsf{\Pi }}}_{1(n-2)}{f}_{\vartheta }({\tau }_1,\epsilon )({\epsilon }^{n-1}{\overline{\vartheta }}_{n-1}({\epsilon }^2{\tau }_1,\epsilon )\\ +{\epsilon }^{n-1}{\mathsf{\Pi }}_{0(n-1)}\vartheta (\epsilon {\tau }_1,\epsilon )+{\Delta }_n\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon ))-\epsilon f_{\vartheta }({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{1(n-2)}\vartheta ({\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1{,\epsilon )\}}_{n-2})({\epsilon }^{n-1}{\mathsf{\Pi }}_{1(n-1)}\vartheta ({\tau }_1)\\ +{\Delta }_n{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon ){))]}_{2n-2}+[{\Delta }_n{\mathsf{\Pi }}_{1}w{({\tau }_1,\epsilon )}^{\prime }({\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-2)}{f}_w{(\epsilon {\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}\\ +{\{\epsilon f_w{({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon )\}}_{n-2}{)]}_{2n-2}\\ +[{\Delta }_n{\mathsf{\Pi }}_{1}u{({\tau }_1,\epsilon )}^{\prime }({\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-2)}{f}_u{(\epsilon {\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}+\{\epsilon f_u({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1{,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon ){\}}_{n-2}{)]}_{2n-2})d{\tau }_1.\end{array}$$

Integrating by parts the first term in the last expression and dropping known terms, we obtain

$$\begin{array}{c}-{[{\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi {(0,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+\frac{1}{\epsilon }{E}_2+E_3){\Delta }_n{\mathsf{\Pi }}_{1}w(0,\epsilon )]}_{2n-2}\\ +{\int }_0^{+\infty }(\underset{\{2\}}{\underbrace{{[{\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi {(\epsilon {\tau }_1,\epsilon )}^{\prime }(\frac{1}{\epsilon }{E}_1+\frac{1}{\epsilon }{E}_2+E_3)\frac{d{\tilde{\mathsf{\Pi }}}_{1(n-1)}w({\tau }_1,\epsilon )}{d{\tau }_1}]}_{2n-2}}}\\ -[{\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi {(\epsilon {\tau }_1,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3)(\epsilon {\widehat{\mathsf{\Pi }}}_{1(n-2)}{f}_{\vartheta }({\tau }_1,\epsilon )({\Delta }_n\overline{\vartheta }({\epsilon }^2{\tau }_1,\epsilon )+{\Delta }_n{\mathsf{\Pi }}_0\vartheta (\epsilon {\tau }_1,\epsilon ))\\ +\epsilon ({\widehat{\mathsf{\Pi }}}_{1(n-2)}{f}_{\vartheta }({\tau }_1,\epsilon )+\underset{\{2\}}{\underbrace{f_{\vartheta }({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon )}}){\epsilon }^{n-1}{\mathsf{\Pi }}_{1(n-1)}\vartheta ({\tau }_1)\\ +\epsilon f_{\vartheta }({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon )+{\tilde{\mathsf{\Pi }}}_{1(n-2)}\vartheta ({\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1,\epsilon ){\Delta }_n{\mathsf{\Pi }}_1\vartheta ({\tau }_1,\epsilon ){)]}_{2n-2}\\ -\underset{\{2\}}{\underbrace{{\mathsf{\Pi }}_{1(n-1)}w{({\tau }_1)}^{\prime }{[A{({\epsilon }^2{\tau }_1)}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon )]}_{n-1}}}\\ -\underset{\{2\}}{\underbrace{{\mathsf{\Pi }}_{1(n-1)}u{({\tau }_1)}^{\prime }{[B{({\epsilon }^2{\tau }_1)}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1,\epsilon )]}_{n-1}}}\\ +[{\Delta }_n{\mathsf{\Pi }}_1\vartheta {({\tau }_1,\epsilon )}^{\prime }({\{\epsilon {\widehat{\mathsf{\Pi }}}_{0(n-2)}{f}_{\vartheta }{(\epsilon {\tau }_1,\epsilon )}^{\prime }{\tilde{\overline{\psi }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\}}_{n-2}+\{\epsilon f_{\vartheta }({\tilde{\overline{\vartheta }}}_{n-2}({\epsilon }^2{\tau }_1,\epsilon )\\ +{\tilde{\mathsf{\Pi }}}_{0(n-2)}\vartheta (\epsilon {\tau }_1,\epsilon ),{\epsilon }^2{\tau }_1{,\epsilon )}^{\prime }(\epsilon E_1+E_2+E_3){\tilde{\mathsf{\Pi }}}_{0(n-2)}\psi (\epsilon {\tau }_1\end{array}$$