## 3 Solution Search Strategy  
We consider the transition to the parametric optimization problem from the control problem (8), i.e., to the problem of finding unknown coefficients of the plant control and the observer control. The plant control constraints of parallelepiped type should be taken into account. To implement this transition, we use the following assumptions:  
1. The set of initial states \(\Omega\) is a parallelepiped, defined by the direct product of segments \([\alpha_{i};\beta_{i}],i=\overline{1,n}\), i.e., \(\Omega=[\alpha_{1};\beta_{1}]\times\cdots\times[\alpha_{n};\beta_{n}]\). With the help of a step \(\Delta x_{i}\), all line segments are divided into \(N_{i}\) segments and the parallelepiped \(\Omega\) is divided into \(N=N_{1}\cdots N_{n}\) elementary disjoint subsets \(\Omega_{k}\), \(k=\overline{1,N}\). In each elementary subset \(\Omega_{k}\), an initial state \(x_{0}^{k}\) (the center of the parallelepiped \(\Omega_{k}\) is specified;
2. The direct product \(Q=[\underline{x_{1}},\overline{x_{1}}]\times\cdots\times[\underline{x_{n}}, \overline{x_{n}}]\) represents the set of admissible values of the state vector coordinates, where \(\underline{x_{i}},\overline{x_{i}},\ i=\overline{1,n}\) are the lower and upper boundaries for each coordinate, respectively, determined by the applied problem being solved. Therefore, one can assume that the possible estimates of the state vector should satisfy the following conditions: \(\hat{x}_{1}\in[\underline{x_{1}},\overline{x_{1}}],\ldots,\hat{x}_{n}\in[ \underline{x_{n}},\overline{x_{n}}]\);
3. The plant control policy is searched in the form: \[u_{j}(t,\hat{x}(t))=\mathrm{sat}\big{\{}g_{j}(t,\hat{x}_{1}(t),\ldots,\hat{x }_{n}(t))\big{\}},j=\overline{1,q},\] (9)where saturation function sat guarantees the fulfillment of the plant control constraints of the form \(a_{j}\leq u_{j}(t)=u_{j}(t,\hat{x}(t))\leq b_{j}\):  
\[\mathsf{sat}\,v_{j}(t)=\left\{ \begin{array}{cc}v_{j}(t),&a_{j}<v_{j}(t)<b_{j}, \\ a_{j},&v_{j}(t)\leq a_{j},\\ b_{j},&v_{j}(t)\geq b_{j},\end{array} \right. \tag{10}\]  
\[\hat{g}_{j}(t,\hat{x}_{1},\ldots,\hat{x}_{n})=\sum_{i_{0}=0}^{L_{0}-1}\sum_{i_{ 1}=0}^{L_{1}-1}\cdots\sum_{i_{n}=0}^{L_{n}-1}u_{i_{0},i_{1},\ldots,i_{n}}^{j} \cdot q(i_{0},t)p_{1}(i_{1},\hat{x}_{1})\cdots p_{n}(i_{n},\hat{x}_{n}), \tag{11}\]  
where \(u_{i_{0},i_{1},\ldots,i_{n}}^{j}\) are unknown coefficients; \(L_{0},L_{1},\ldots,L_{n}\) are scales of truncation; \(q(i_{0},t),i_{0}=\overline{0,L_{0}-1}\) is a system of orthonormal time functions (basis functions) defined on the segment \([t_{0},t_{1}]\) and satisfying the condition \(\int\limits_{t_{0}}^{t_{1}}q(i,t)q(j,t)dt=\left\{ \begin{array}{cc}1,i=j,&; \text{ and }p_{j}(i_{j},\hat{x}_{j}),i_{j}=\overline{0,L_{j}-1}\end{array} \right.\) is a system of orthonormal functions of a variable \(\hat{x}_{j}\) (basis functions) defined on an interval \([\underline{x_{j},\overline{x_{j}}}],j=1,\ldots,n\). As the basis functions \(q(i_{0},t),\,p_{k}(i_{k},\hat{x}_{k}),\,k=\overline{1,n}\), one can take, for example:  
* Legendre polynomials: \[p(n,x)=\sum_{k=0}^{n}\left(C_{n}^{k}\right)^{2}\widetilde{x}^{n-k}\,\tilde{x} ^{k},\,n=\overline{0,L-1};\]
* Cosine: \[p(n,x)=\cos(n\cdot\pi\cdot(2\cdot\widetilde{x}-1)),n=\overline{0,L-1};\]  
where \(\widetilde{x}=(x-\underline{x})/(\overline{x}-\underline{x})\) and other systems of basic functions. The matrix entries \(K_{ij}(t,\hat{x}),i=1,\ldots,n;j=1,\ldots,m\) of the state observer control policy \(K(t,\hat{x})\) are found by a formula similar to (11), where variable \(u\) is replaced by \(K\). The value of the pencil control cost functional (7) is approximated as:  
\[J[u(t,\hat{x}),\,K(t,\hat{x})]\cong(1/N)\sum_{k=1}^{N}I(x_{0},u(t,\hat{x}(t)), K(t,\hat{x}(t))). \tag{12}\]  
The optimization problem is to choose the best parameters \(u_{i_{0},i_{1},\ldots,i_{n}}^{j},\,K_{i_{0},i_{1},\ldots,i_{n}}^{j,j}\), minimizing performance index (12) by using a mini-batch adaptive method of random search (MAMRS) [4].