## 2 Preliminaries and Problem Formulation  
In this section, we give a review of the basic terms that include nonlinear state-feedback \(\mathcal{H}_{\infty}\) control and related differential game theory, and this is mostly based on classic references from this field [1; 2; 3; 4; 21; 22]. Consider the causal input-affine nonlinear system in the state space defined on a some manifold \(\mathcal{X}\subseteq\mathbb{R}^{n}\) in the following form  
\[\Sigma:\quad \begin{cases}\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})+\mathbf{G}_{ 1}(\mathbf{x})\mathbf{u}+\mathbf{G}_{2}(\mathbf{x})\mathbf{d},\ \ \mathbf{x}(t_{0})=\mathbf{x}_{0},\\ \mathbf{y}=\mathbf{x},\\ \mathbf{z}=\mathbf{h}(\mathbf{x})+\mathbf{L}(\mathbf{x})\mathbf{u},\end{cases}  \tag{1}\]  
where \(\mathbf{x}\in\mathcal{X}\) is the state vector, \(\mathbf{u}\in\mathcal{U}\subseteq\mathbb{R}^{m}\) is the control vector, \(\mathbf{d}\) is the vector representing internal/external uncertainty belonging to the set \(\mathcal{D}\subset\mathcal{L}_{2}\Big{(}\Big{[}t_{0},\,t_{f}\Big{]},\,\mathbb{R }^{s}\Big{)}\). The output vector \(\mathbf{y}\in\mathbb{R}^{n}\) contains all directly measured states of system \(\Sigma\). The vector \(\mathbf{z}\in\mathbb{R}^{q}\) is the performance variable. Furthermore, the functions \(\mathbf{f}:\mathcal{X}\rightarrow\mathcal{X}\), \(\mathbf{G}_{1}:\mathcal{X}\rightarrow\mathcal{M}^{n\times m}(\mathcal{X})\), \(\mathbf{G}_{2}:\mathcal{X}\rightarrow\mathcal{M}^{n\times s}(\mathcal{X})\), \(\mathbf{h}:\mathcal{X}\rightarrow\mathbb{R}^{q}\), \(\mathbf{L}:\mathcal{X}\rightarrow\mathcal{M}^{q\times m}(\mathcal{X})\) are real \(C^{1}\)-functions of \(\mathbf{x}\). The following assumptions are employed:  
**Assumption 1**. : \(\mathbf{x}=\mathbf{0}\) _is a unique equilibrium point, with \(\mathbf{u}=\mathbf{0}\) and \(\mathbf{d}=\mathbf{0}\), of system \(\Sigma\) and for simplicity \(\mathbf{f}(\mathbf{0})=\mathbf{0}\), \(\mathbf{h}(\mathbf{0})=\mathbf{0}\)._  
**Assumption 2**. : _The vector \(\mathbf{h}(\mathbf{x})\) and matrix \(\mathbf{L}(\mathbf{x})\) are such that \(\mathbf{h}(\mathbf{x})^{\mathrm{T}}\mathbf{L}(\mathbf{x})=\mathbf{0}\ i\, \mathbf{L}(\mathbf{x})^{\mathrm{T}}\mathbf{L}(\mathbf{x})=\mathbf{I}\) form all \(\mathbf{x}\in\mathcal{X}\), which implies_  
\[\mathbf{z}= \begin{bmatrix}\mathbf{h}(\mathbf{x})\\ \mathbf{u}\end{bmatrix} \Longrightarrow\|\mathbf{z}\|^{2}=\|\mathbf{h}(\mathbf{ x})\|^{2}+\|\mathbf{u}\|^{2}. \tag{2}\]  
By introducing Assumption 2, the so-called singular problem is avoided. More details on solving the singular problem can be found in [23; 24]. **Assumption 3**. : _Initial state vector \(\mathbf{x}_{0}\) is a priori known._  
Furthermore, the following definition is introduced:  
**Definition 1** (\(\mathcal{L}_{2}\)-gain). : \(\mathcal{L}_{2}\)_-gain from \(\mathbf{d}\) to \(\mathbf{z}\) of system \(\Sigma\) is the supremum of \(\gamma>0\) satisfies_  
\[\|\mathbf{z}\|_{\mathcal{L}_{2}}^{2}\leq\gamma^{2}\|\mathbf{d}\|_{\mathcal{L} _{2}}^{2}+\beta(\mathbf{x}_{0}), \tag{3}\]  
_for some bounded \(C^{0}\)-function \(\beta:U\subset\mathcal{X}\rightarrow\mathbb{R}\) such that \(\beta(\mathbf{0})=0\)._In general, the problem of nonlinear \(\mathcal{H}_{\infty}\) control of system \(\Sigma\) where all state variables are available (measurable or can be estimated) can be formulated as follows:  
**Problem 1**. : _The problem of optimal nonlinear state-feedback \(\mathcal{H}_{\infty}\) control of system \(\Sigma\) is to determine the control law \(\mathbf{u}^{*}=\boldsymbol{\mu}(\mathbf{x},t)\) and the \"worst case\" \(\mathbf{d}^{*}=\boldsymbol{\
u}(\mathbf{x},t)\) such that \(\gamma>0\) is minimized._  
**Assumption 4**. : _The functions \(\boldsymbol{\mu}(\mathbf{x},t)\) and \(\boldsymbol{\
u}(\mathbf{x},t)\) are \(\boldsymbol{\mu}\in C^{1}(\mathcal{X})\), \(\boldsymbol{\
u}\in C^{1}(\mathcal{X})\)._  
If for the system \(\Sigma\) there exists some \(\gamma\geq\gamma^{*}\) that satisfies (3), then a zero-sum differential game can be defined.