## 4 The Control Correction  
Let us represent \(x(t,\varepsilon),u(t,\varepsilon)\) in the form of \(\varepsilon\) integer powers series and substitute them in the conditions of problem (1), then in addition to (2) in the first approximation we obtain the following control problem with fixed ends  
\[\dot{x}_{1}=F_{0,x}(t,x_{0}(t),u_{0}(t))x_{1}(t)+F_{0,u}(t,x_{0}(t),u_{0}(t)) u_{1}(t)+F_{1}(t,x_{0}(t),u_{0}(t)), \tag{12}\]  
\(x_{1}(t_{0})=0,\quad x_{1}(t_{f})=0\),  
where \(F_{0,x}=\frac{\partial F_{0}}{\partial x}\), \(F_{0,u}=\frac{\partial F_{0}}{\partial u}\). Here, the unperturbed system is assumed to be flat and the control \(u_{0}(t)\) is found using the covering method. The first (main) part of the control correction \(u_{1}(t,\varepsilon)\) will be sought in the form \(u_{1}(t,\varepsilon)=\varepsilon u_{1}(t)\). The term \(\omega(t)\) is intended to clarify the solution in the original perturbed problem. The definition of \(u_{1}(t)\) leads to the solution of the Fredholm equation of the first kind. Therefore, determining \(x_{0}(t),u_{0}(t)\) from problem (2), let us find \(u_{1}(t)\) from (12) now, assuming \(\omega(t)=0\). We have  
\[0=x_{1}(t_{f})=\Phi(t_{f})\int\limits_{t_{0}}^{t_{f}}\Big{(}\Phi^{-1}(\tau)(F_ {0,u}(\tau,x_{0}(\tau),u_{0}(\tau))u_{1}(\tau)+F_{1}(\tau,x_{0}(\tau),u_{0}( \tau)))\Big{)}d\tau,\]  
where \(\Phi(t)\) is a fundamental matrix for a linear system \(\dot{y}=F_{0,x}(x_{0}(t),u_{0}(t))y\), i.e., \(\Phi=F_{0,x}(x_{0}(t),u_{0}(t))\Phi\), \(\Phi(t_{0})=I\). Given that \(\Phi(t)\) is a non-degenerate matrix for any \(t\), we obtain the next Fredholm matrix integral equation of the first kind  
\[\int\limits_{t_{0}}^{t_{f}}K(\tau)u_{1}(\tau)d\tau=b, \tag{13}\]where  
\[K(\tau)=\Phi^{-1}(\tau)F_{0,u}(\tau,x_{0}(\tau),u_{0}(\tau)),\quad b=-\int\limits_ {l_{0}}^{t_{f}}\Bigl{(}\Phi^{-1}(\tau)F_{1}(\tau,x_{0}(\tau),u_{0}(\tau))\Bigr{)} d\tau.\]  
It is known that equations of this type are incorrectly posed that means small changes in the conditions of the problem caused by the approximate nature of solution of (13), that occur during computation may lead to non-uniqueness of the solution and significant changes in the obtained \(u_{1}\) value. We use the [PERSON]'s regularization [12] (see also in [13]), according to which the approximate solution in (13) is found as a vector minimizing the next functional  
\[ \begin{split} M^{\alpha}(u_{1})&=\left(\int\limits_ {l_{0}}^{t_{f}}K(\tau)u_{1}(\tau)d\tau-b\right)^{\prime}\left(\int\limits_{l_{ 0}}^{t_{f}}K(\tau)u_{1}(\tau)d\tau-b\right)+\\ &\quad\alpha\int\limits_{l_{0}}^{t_{f}}(\dot{u}_{1}^{\prime}(\tau )P(\tau)\dot{u}_{1}(\tau)+u_{1}^{\prime}(\tau)Q(\tau)u_{1}(\tau)\Bigr{)}d\tau, \end{split}  \tag{14}\]  
where \(\alpha>0\) is a regularization parameter, sign \" / \" stands for transposition, and \(P\) and \(Q\) are positive definite weight matrices. Of course, here \(\alpha\) should be significantly smaller than \(\varepsilon\), for example, \(\alpha=O(\varepsilon^{k})\), \(k\geq 2\). Here, we will find the solution of problem (14) approximately, using an auxiliary nonlinear programming problem, for example, assuming that each coordinate in \(u_{1}\) is represented using a series of orthogonal functions, so that  
\[u_{1,j}(\tau(t))=\frac{1}{2}c_{i,0}+\sum\limits_{j=1}^{k_{u}}\bigl{(}c_{i,2j-1 }\cos(j\tau)+c_{i,2j}\sin(j\tau)\bigr{)},\quad i=1,\ldots,r, \tag{15}\]  
where  
\[\tau=-\frac{2\pi t}{t_{0}-t_{f}}+\frac{\pi(t_{0}+t_{f})}{t_{0}-t_{f}},\quad t \in[t_{0},t_{f}],\quad\tau\in[-\pi,\pi],\]  
and \(c_{l}\) are the vectors of unknown coefficients for each vector function \(u_{1}(t)\) coordinate, \(l=0,1,\ldots,2k_{u}\); the \(k_{u}\geq 0\) is a parameter determines the number of terms in the series. Thus, the coefficients \(c_{i,l}(\alpha)\) are found using an iterative algorithm for solving the nonlinear programming problem  
\[M^{\alpha}\to\min_{c_{i,l}},\]  
where the next operations are made at each step \(p\), so the scheme of algorithm have the next form:  
1.