## 3 Synthesis of the Algorithm for the Solution of the Zero-Sum Differential Game
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**Algorithm 1** quasi-Newton method for \(\mathcal{L}_{2}\)-gain minimization. To solve the subproblem in the second step of Algorithm 1, we use the conjugate gradient method as described in the following algorithm. Note that, in Algorithm 2, the maximization of the objective function is obtained by changing the sign in front of the gradient with respect to \(\rho\) in (23). Additionally, it should be noted that the strategy for solving Problem 2 proposed by Algorithms 1 and 2 requires appropriate initialization, i.e., appropriate selection of approximation functions and initial parameters \(\boldsymbol{\pi}_{0}\), \(\boldsymbol{\rho}_{0}\) and \(\mu_{0}\). If they are inadequate, it cannot be guaranteed that the control law is for the \"worst case\" of uncertainty. In the general, it is very difficult to guarantee whether a solution for the \"worst case\" of uncertainty is obtained. However, the conditions that would give recommendations for the initialization of the parameters of the proposedalgorithms could be derived by applying the concept of inverse min max optimal control. Solving the problem of inverse min-max optimal control goes beyond the scope of the research presented in this paper and will be part of future research. ```
0:\(\mathbf{x}_{0}\in\mathbb{R}^{n}\), \(\boldsymbol{\xi}_{0}\equiv\left[\pi_{0}^{\text{T}}\ \boldsymbol{\rho}_{0}^{\text{T}}\right]^{\text{T}}\in\mathbb{R}^{n_{\ell}}\), \(\mu_{k}\), \(\beta_{0}\in\mathbb{R}_{+}\), \(\eta_{0}\in\mathbb{R}_{+}\), \(\epsilon\in\mathbb{R}_{+}\). 0:\(\boldsymbol{\pi}^{\star}\), \(\boldsymbol{\rho}^{\star}\). 1: Set \(j\gets 1\). 2: Perform a conjugate gradient descent/ascent algorithm in the following form \[\boldsymbol{\xi}_{j} =\boldsymbol{\xi}_{j-1}+\eta_{j-1}\mathbf{s}_{j-1},\] (21) \[\mathbf{s}_{j} =-\mathcal{J}(\boldsymbol{\xi}_{j})+\beta_{j-1}\mathbf{s}_{j-1},\] (22) where \(\mathbf{s}\) is the search direction vector and \[\mathcal{J}(\boldsymbol{\xi}_{j}) =\left[\
abla_{\boldsymbol{\pi}}J-\
abla_{\boldsymbol{\rho}} \right]^{\text{T}},\] (23) \[J =\|\mathbf{h}(\mathbf{x}_{j})\|_{\mathcal{L}_{2}}^{2}+\|\boldsymbol {\Theta}(\mathbf{x}_{j})\boldsymbol{\pi}_{j}\|_{\mathcal{L}_{2}}^{2}-\mu_{k} \|\boldsymbol{\
abla}(\mathbf{x}_{j})\boldsymbol{\rho}_{j}\|_{\mathcal{L}_{2 }}^{2}.\] (24)
3: Determine \(\eta_{j}>0\) by applying the line search strategy that satisfies [PERSON]'s conditions (see Algorithm 3). 4: Determine \(\beta_{j}>0\) by applying Dai-Yuan method [37]. 5: If \(\|\mathcal{J}(\boldsymbol{\xi}_{j})\|_{\infty}\leq\epsilon\) then terminate. Otherwise, set \(j\gets j+1\) and return to step 2. ```  
**Algorithm 2** Conjugate gradient method for saddle point computation. ```
0:\(\boldsymbol{\xi}_{j}\), \(\mathbf{s}_{j}\), \(J(\boldsymbol{\xi}_{j})\), \(0<c_{1}<c_{2}<1\), \(v\in(0,\ 1)\). 0:\(\eta_{j}\). 1: Set \(l\gets 0\). 2: Choose initial \(\eta_{0}\). 3: While \(\eta_{l}\) satisfies [PERSON]'s conditions \[J(\boldsymbol{\xi}_{j}+\eta_{l}\mathbf{s}_{j}) \leq J(\boldsymbol{\xi}_{j})+c_{1}\eta_{l}\mathcal{J}(\boldsymbol{\xi}_{j} )\mathbf{s}_{j},\] (25) \[\mathcal{J}(\boldsymbol{\xi}_{j}+\eta_{l}\mathbf{s}_{j})\mathbf{s}_{j} \geq c_{2}\mathcal{J}(\boldsymbol{\xi}_{j})\mathbf{s}_{j},\] (26) calculate (i) \(\eta_{l+1}=v\eta_{l}\), (ii) \(l\gets l+1\). 4: Set \(\eta_{j}=\eta_{l}\). ```  
**Algorithm 3** Line search method with Wolfe conditions. Furthermore, the initialization of parameters \(\beta\) and \(\eta\) is dependent on a specific optimization problem. Through a series of numerical experiments in simulations, we determined that the Dai-Yuan conjugate gradient method reaches a similar level of numerical robustness and accuracy for various initial values of \(\beta_{0}\), whereas for the standard gradient algorithm, choosing the initial parameters largely affects the convergence and can cause numerical instabilities. To determine the convergence step \(\eta_{j}>0\), we use a line search method that satisfies the Wolfe conditions as described in the following algorithm.