## 3 Covering Method  
Here, we represent the covering method used to find an open-loop control for the unperturbed system. Let \(\mathcal{X}\) and \(\mathcal{Y}\) be two given dynamical systems. A surjective mapping \(\
u\) of the extended phase space of the system \(\mathcal{X}\) into the extended phase space of the system \(\mathcal{Y}\) is called a _covering_ of \(\mathcal{X}\) into \(\mathcal{Y}\) if the following conditions hold:  
1. the mapping \(\
u\) is regular, i.e., at any point rank of differential \(d\
u\) is equal to the dimension of the image;
2. the mapping \(\
u\) maps each trajectory of the system \(\mathcal{X}\) into a trajectory of the system \(\mathcal{Y}\);
3. the mapping of trajectories is regular, i.e., at any point tangent vector to a trajectory does not belong to kernel of \(d\
u\). From the definition of covering it follows that the preimage of any trajectory of \(\mathcal{Y}\) consists of points of trajectories of some subsystem of \(\mathcal{X}\). Let us consider the solution of the point-to-point problem using covering method. Let system  
\[\dot{x}=f(t,x,u),\qquad x\in\mathbb{R}^{n},\quad u\in\mathbb{R}^{m}, \tag{3}\]  
be given, \(\mathcal{O}\) be a domain of the space with coordinates \(x,u,\dot{u},\ldots,u^{(k)}\), \(S_{0}\subset\mathcal{O}\) an initial set, \(S_{f}\subset\mathcal{O}\) a target set. For a pair of vector functions \(x(t),u(t)\) and a number \(t_{1}\) denote by \(j_{k}(x(t_{1}),u(t_{1}))\) the point \(x(t_{1}),u(t_{1}),\dot{u}(t_{1}),\ldots,u^{(k)}(t_{1})\) of this space. Knowing the final time \(t_{f}\in(t_{0},\infty)\), the problem is to compute an admissible input signal \(u\colon\ [t_{0},t_{f}]\to\mathbb{R}^{m}\) such that the system trajectory satisfies  
\[\forall t\in[t_{0},t_{f}]\qquad j_{k}(x(t),u(t))\in\mathcal{O}\] \[j(x_{k}(t_{0}),u(t_{0}))\in S_{0},\quad j_{k}(x(t_{f}),u(t_{f}) )\in S_{f}. \tag{4}\]  
Let \(g_{1}\colon\ \mathcal{O}\to\mathbb{R}^{n_{1}}\) and \(g_{2}\colon\ \mathcal{O}\to\mathbb{R}^{n_{2}}\) be two mappings. Consider the case when the initial set \(S_{0}\) and the target set \(S_{f}\) are determined by the equalities  
\[S_{0} \colon\ g_{1}(x,u,\dot{u},\ldots,u^{(k)})=g_{1,0}, \tag{5}\] \[S_{f} \colon\ g_{2}(x,u,\dot{u},\ldots,u^{(k)})=g_{2,f}. \tag{6}\]  
Suppose functions \(U_{i}\) and \(\phi_{j},i=\overline{1,m},j=\overline{1,n_{2}}\), of the variables  
\[x_{1},\ldots,x_{n},u_{1},\dot{u}_{1},\ldots,u_{1}^{(k_{1}-1)},u _{2},\ldots,u_{m}^{(k_{m}-1)},\] \[\forall i\quad k_{i}\leq k,\quad k_{1}+\ldots+k_{m}+n=n_{1}+n_{2}\]  
satisfy the following conditions. (A) The relations \(p_{j}=\phi_{j},j=\overline{1,n}\), define a covering from the system  
\[\dot{x}_{j} = f_{j}(t,x_{1},\ldots,u_{m}),\quad j=\overline{1,n}, \tag{7}\] \[u_{i}^{(k_{i})} = U_{i}(t,x_{1},\ldots,u_{m}^{(k_{m}-1)}),\qquad i=\overline{1,m}, \tag{8}\]into a system of the form  
\[\dot{p}=P(t,p),\quad p\in\mathbb{R}^{n_{2}}. \tag{9}\]  
(B) The given final value \(g_{2,f}\) uniquely determines the value \(p_{f}=p(t_{f})\). Conversely, the value \(p(t_{f})\) uniquely determines the value \(g_{2,f}\). (C) If \(p(t)\) is the solution of system (9) satisfying the condition \(p(t_{f})=p_{f}\), then the nonlinear vector equation  
\[\forall t\in[t_{0},t_{f}]\qquad j_{k}(x(t),u(t))\in\mathcal{O}\]0  
is solvable for \(u_{1}(t_{0}),\ldots,u_{m}^{(k_{m}-1)}(t_{0})\). If conditions (A), (B), and (C) hold, then the problem (3)-(6) can be solved as follows. 1. The value \(p(t_{f})\) is found from the final conditions (6). 2. One finds the solution \(p(t)\) of system (9) satisfying the condition \(p(t_{f})=p_{f}\) (that is, the solution of the Cauchy problem in reverse time, from \(t_{f}\) to \(t_{0}\)). 3. One computes \(p_{0}=p(t_{0})\). 4. The values \(u_{1}(t_{0}),\dot{u}_{1}(t_{0}),\ldots,u_{m}^{(k_{m}-1)}(t_{0})\) are found from system (10). 5. By solving the Cauchy problem for system (7) and (8) with the initial values \[\forall t\in[t_{0},t_{f}]\qquad j_{k}(x(t),u(t))\in\mathcal{O}\]1 (11)  
one finds a solution \(\big{(}x(t),u(t)\big{)}\) of system (3).