## 3 Synthesis of the Algorithm for the Solution of the Zero-Sum Differential Game  
In this section, an approach of determining control and uncertainty variables for optimal \(\mathcal{H}_{\infty}\) control (Problem 1) of nonlinear dynamic system \(\Sigma\) is proposed. The proposed approach does not require solving the HJI Equation (7), but the solution is reduced to the direct numerical determination of the saddle point of the related zero-sum differential game. Control and uncertainty variables are approximated by functions with a linear dependence on a finite number of constant parameters. To calculate these parameters, an approach based on the integration of the quasi-Newton method, the conjugate gradient method, the Adams method and the complex-step derivative approximation into one algorithm is proposed. The goal is to obtain a control variable that explicitly depends on the state variables and in that form is simple for practical implementation. Additionally, the aim is to achieve a numerical solution that uniformly converges towards the optimal solution by increasing the order of complexity of the approximation, i.e., by increasing the number of parameters. Since we introduced the Assumption 4, based on [PERSON]'s theorem ([25], p. 65) (which refers to polynomial approximation functions) and its generalizations [26; 27; 28; 29] (which refer to non-polynomial forms of nonlinear approximation functions) on the uniform approximation of smooth functions, there are constants \(p_{j}^{i},r_{j}^{i}\in\mathbb{R}\) such that the \(i\)-th component of control and uncertainty vector can be written in the following form:  
\[\hat{a}_{i}(\mathbf{x}) =\sum_{j=1}^{n_{\theta}}p_{j}^{i}\theta_{j}^{i}(\mathbf{x}), \tag{8}\] \[\hat{d}_{i}(\mathbf{x}) =\sum_{j=1}^{n_{\theta}}r_{j}^{i}\psi_{j}^{i}(\mathbf{x}), \tag{9}\]  
where \(\theta_{j}^{i}(\mathbf{x})\in C^{1}(\mathcal{X})\), \(\psi_{j}^{i}(\mathbf{x})\in C^{1}(\mathcal{X})\) such that \(\theta_{j}^{i}(\mathbf{0})=0\), \(\psi_{j}^{i}(\mathbf{0})=0\). Linear subspaces generated by sets \(\{\theta_{j}^{i}(\mathbf{x})\}\) i \(\{\psi_{j}^{i}(\mathbf{x})\}\) are dense in the Sobolev norm \(W^{1,\infty}\)[30]. For well chosen functions \(\theta_{j}^{i}(\mathbf{x})\) and \(\psi_{j}^{i}(\mathbf{x})\), we have  
\[|\hat{a}_{i}(\mathbf{x})-u_{i}(\mathbf{x})| <\varepsilon_{u_{i}}(\mathbf{x}), \tag{10}\] \[|\hat{d}_{i}(\mathbf{x})-d_{i}(\mathbf{x})| <\varepsilon_{d_{i}}(\mathbf{x}), \tag{11}\]  
where \(\varepsilon_{u_{i}}(\mathbf{x})\) i \(\varepsilon_{d_{i}}(\mathbf{x})\) are the approximation errors. It follows that \(\varepsilon_{u_{i}}(\mathbf{x})\to 0\), \(\varepsilon_{d_{i}}(\mathbf{x})\to 0\) when \(n_{\theta}\to\infty\), \(n_{\psi}\to\infty\), respectively, while for fixed \(n_{\theta}\) and \(n_{\psi}\), approximation errors are bounded by constants on the compact set. It is well known from approximation theory (see for example [31; 32]) that it is often possible to determine in advance how many terms of the basis functions expansion should be taken for the required accuracy. Equations (8) and (9) can be written in the following matrix form:  
\[\hat{\mathbf{u}}(\mathbf{x}) =\mathbf{\Theta}(\mathbf{x})\,\boldsymbol{\pi}, \tag{12}\] \[\hat{\mathbf{d}}(\mathbf{x}) =\mathbf{\Psi}(\mathbf{x})\,\boldsymbol{\rho}, \tag{13}\]  
where  
\[\mathbf{\Theta}(\mathbf{x}) \equiv \begin{bmatrix}\theta^{1}(\mathbf{x})&\mathbf{0}&\cdots& \mathbf{0}\\ \mathbf{0}&\theta^{2}(\mathbf{x})&\cdots&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{0}&\mathbf{0}&\cdots&\theta^{m}(\mathbf{x})\end{bmatrix} ,\quad \mathbf{\Psi}(\mathbf{x})\equiv \begin{bmatrix}\psi^{1}(\mathbf{x})&\mathbf{0 }&\cdots&\mathbf{0}\\ \mathbf{0}&\psi^{2}(\mathbf{x})&\cdots&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{0}&\mathbf{0}&\cdots&\psi^{5}(\mathbf{x})\end{bmatrix} ,\] \[\boldsymbol{\theta}^{i}(\mathbf{x}) \equiv \begin{bmatrix}\theta_{1}^{i}(\mathbf{x})&\theta_{2}^{i} (\mathbf{x})&\cdots&\theta_{n_{\psi}}^{i}(\mathbf{x})\end{bmatrix} ,\quad \psi^{i}(\mathbf{x})\equiv