## 3 Synthesis of the Algorithm for the Solution of the Zero-Sum Differential Game
\begin{bmatrix}\psi_{1}^{i}(\mathbf{x})&\psi_{2}^{i }(\mathbf{x})&\cdots&\psi_{n_{\psi}}^{i}(\mathbf{x})\end{bmatrix} , \tag{14}\] \[\boldsymbol{\pi} \equiv \begin{bmatrix}\mathbf{p}_{2}^{1}\\ \mathbf{p}_{2}^{2}\\ \vdots\\ \mathbf{p}^{m}\end{bmatrix} ,\quad\boldsymbol{\rho}\equiv \begin{bmatrix}r_{2 }^{1}\\ r_{2}^{2}\\ \vdots\\ r_{n_{\psi}}^{i}\end{bmatrix} ,\quad\mathbf{r}^{i}\equiv \begin{bmatrix}r_{1 }^{i}\\ r_{2}^{i}\\ \vdots\\ r_{n_{\psi}}^{i}\end{bmatrix} .\]  
In this paper, based on the previous considerations, the problem that needs to be solved can be formulated as follows:  
**Problem 2**. : _Determine the parameters \(\boldsymbol{\pi}\) and \(\boldsymbol{\rho}\) such that the \(\mathcal{L}_{2}\)-gain of the closed-loop_  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})+\mathbf{G}_{1}(\mathbf{x})\mathbf{ \Theta}(\mathbf{x})\boldsymbol{\pi}+\mathbf{G}_{2}(\mathbf{x})\mathbf{\Psi}( \mathbf{x})\boldsymbol{\rho}, \tag{15}\]is minimized. In other words, according to Definition 1, solve the following minimax optimization problem_  
\[J_{\mu}^{*}(\mathbf{x}_{0})=\min_{\pi}\max_{\rho}\Big{\{}\|\mathbf{h}(\mathbf{x} )\|_{\mathcal{L}_{2}}^{2}+\|\mathbf{\Theta}(\mathbf{x})\boldsymbol{\pi}\|_{ \mathcal{L}_{2}}^{2}-\mu\|\boldsymbol{\Psi}(\mathbf{x})\boldsymbol{\rho}\|_{ \mathcal{L}_{2}}^{2}\Big{\}}, \tag{16}\]  
_which represents a zero-sum differential game, where the minimal \(\mathcal{L}_{2}\)-gain is \(\gamma^{*}=\sqrt{\mu}\)._  
First, to propose an algorithm for minimization of the parameter \(\mu\) in (16), the first-order derivative of the value of the differential game with respect to \(\mu\) is needed. Since \(J_{\mu}^{*}(\mathbf{x}_{0})\) is non-differentiable because it is defined with the min-max operator, the sub-differential is employed. It can easily be shown that sub-differential of \(J_{\mu}^{*}(\mathbf{x}_{0})\) with respect to \(\mu\) is  
\[\partial J_{\mu}^{*}=-\|\boldsymbol{\Psi}(\mathbf{x})\boldsymbol{\rho}\|_{ \mathcal{L}_{2}}^{2}. \tag{17}\]  
Next, due to the inaccurate calculation of the previously mentioned derivative, the quasi-Newton method is considered, for which the superlinear convergence and numerical robustness have been proven (see for example [33; 34; 35; 36]). The quasi-Newton \(k\)-th iteration is defined by  
\[ \begin{split}\mu_{k+1}&=\mu_{k}-\frac{\|\mathbf{h}( \mathbf{x}_{k})\|_{\mathcal{L}_{2}}^{2}+\|\mathbf{\Theta}(\mathbf{x}_{k}) \boldsymbol{\pi}_{k}\|_{\mathcal{L}_{2}}^{2}-\mu_{k}\|\boldsymbol{\Psi}( \mathbf{x}_{k})\boldsymbol{\rho}_{k}\|_{\mathcal{L}_{2}}^{2}}{\partial J_{\mu _{k}}^{*}}\\ &=\frac{\|\mathbf{h}(\mathbf{x}_{k})\|_{\mathcal{L}_{2}}^{2}+\| \mathbf{\Theta}(\mathbf{x}_{k})\boldsymbol{\pi}_{k}\|_{\mathcal{L}_{2}}^{2}} {\|\boldsymbol{\Psi}(\mathbf{x}_{k})\boldsymbol{\rho}_{k}\|_{\mathcal{L}_{2}} ^{2}}.\end{split}  \tag{18}\]  
To minimize the \(\mathcal{L}_{2}\)-gain of the system (15), we propose the following quasi-Newton algorithm. It is important to note that in the second step of Algorithm 1, unlike other known methods of nonlinear optimization, the dynamics of the system (15) is not included in the objective function. ```
0:\(\mu_{0}\in\mathbb{R}_{+},\epsilon\in\mathbb{R}_{+}\). 0: Set \(k\gets 0\). 2: Solve zero-sum differential game \[J_{\mu_{k}}^{*}=\min_{\pi}\max_{\boldsymbol{\rho}}\Big{\{}\|\mathbf{h}( \mathbf{x}_{k})\|_{\mathcal{L}_{2}}^{2}+\|\mathbf{\Theta}(\mathbf{x}_{k}) \boldsymbol{\pi}_{k}\|_{\mathcal{L}_{2}}^{2}-\mu_{k}\|\boldsymbol{\Psi}( \mathbf{x}_{k})\boldsymbol{\rho}_{k}\|_{\mathcal{L}_{2}}^{2}\Big{\}},\] (19) where \(\mathbf{x}_{k}\), \(\boldsymbol{\pi}_{k}\) and \(\boldsymbol{\rho}_{k}\) are coupled by (15). 3: Calculate \[\mu_{k+1}=\frac{\|\mathbf{h}(\mathbf{x}_{k})\|_{\mathcal{L}_{2}}^{2}+\| \mathbf{\Theta}(\mathbf{x}_{k})\boldsymbol{\pi}_{k}\|_{\mathcal{L}_{2}}^{2}}{ \|\boldsymbol{\Psi}(\mathbf{x}_{k})\boldsymbol{\rho}_{k}\|_{\mathcal{L}_{2}} ^{2}}.\] (20)
4: If \(|\mu_{k+1}-\mu_{k}|\leq\epsilon\) then terminate. Otherwise, set \(k\gets k+1\) and return to step 2.