## 2 SDRE Method  
Let Equation (1) be the continuous time nonlinear system  
\[\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})+\mathbf{B}\mathbf{u}, \tag{1}\]  
where \(\mathbf{x}\in\mathbb{R}^{n}\) and \(\mathbf{u}\in\mathbb{R}^{m}\) are the state and input vectors, respectively. \(\mathbf{F}(\mathbf{x})\) is a continuous (\(C^{k}\) class) nonlinear function of \(\mathbf{x}\), and \(\mathbf{B}\in\mathbb{R}^{n\times m}\) is the constant control matrix. The problem consists of finding an admissible control \(\mathbf{u}\in\mathbb{R}^{m}\) for \(t_{0}\leq t\), which minimizes the performance index [28; 29; 30; 31]  
\[J(\mathbf{u})=\frac{1}{2}\int\limits_{0}^{\infty}(\mathbf{x}^{T}\mathbf{Q} \mathbf{x}+\mathbf{u}^{T}\mathbf{R}\mathbf{u})dt, \tag{2}\]  
where \(\mathbf{Q}\in\mathbb{R}^{n\times n}\) is a symmetric positive semidefinite, and \(\mathbf{R}\in\mathbb{R}^{m\times m}\) is a symmetric positive-definite matrix. Rewriting the nonlinear Equation (1), the state-dependent coefficient form (SDC) follows  
\[\dot{\mathbf{x}}=\mathbf{A}(\mathbf{x})\mathbf{x}+\mathbf{B}\mathbf{u}, \tag{3}\]  
where \(\mathbf{A}(\mathbf{x})\in\mathbb{R}^{n\times n}\). There are many methods for SDC parameterization. Each parameterization should be true for all \(0\leq\alpha\leq 1\)[32]  
\[\alpha\mathbf{A}_{I}(\mathbf{x})\mathbf{x}+(1-\alpha)\mathbf{A}_{II}(\mathbf{ x})\mathbf{x}=\alpha\mathbf{F}(\mathbf{x})+(1-\alpha)\mathbf{F}(\mathbf{x})= \mathbf{F}(\mathbf{x}). \tag{4}\]  
The control problem for the nonlinear continuous-time system Equation (1) can be formulated as follows. Given nonlinear functions \(\mathbf{F}(\mathbf{x})\in\mathbb{R}^{n}\), \(\mathbf{B}\in\mathbb{R}^{n\times m}\), \(\mathbf{Q}\in\mathbb{R}^{n\times n}\), and \(\mathbf{R}\in\mathbb{R}^{m\times m}\), we attempt to find a control vector \(\mathbf{u}\in\mathbb{R}^{m}\) for \(t\in[t_{0},\infty]\) that allows us to control the system state vector from \(\mathbf{x}_{0}\) to \(\mathbf{x}_{\infty}\) while the performance index is minimized (Equation (19)). If the integral functions \(\mathbf{x}^{T}\mathbf{Q}\mathbf{x}+\mathbf{u}^{T}\mathbf{R}\mathbf{u}\) and \(\mathbf{A}(\mathbf{x})\mathbf{x}+\mathbf{B}\mathbf{u}\) are continuously differentiable functions of each of their arguments, then we may imply that \(\mathbf{u}\in C[t_{0},\infty]\) is a control that minimizes the functional \(J(\mathbf{u}):C[t_{0},\infty]\rightarrow\mathbb{R}\). To solve the problem, consider the Hamiltonian  
\[\mathbf{H}=\frac{1}{2}(\mathbf{x}^{T}\mathbf{Q}\mathbf{x}+\mathbf{u}^{T} \mathbf{R}\mathbf{u})+\mathbf{p}^{T}(\mathbf{A}(\mathbf{x})\mathbf{x}+ \mathbf{B}\mathbf{u}), \tag{5}\]  
where \(\mathbf{p}\in\mathbb{R}^{n}\) is the co-state vector. To be consistent with the Pontryagin minimum principle, if \(\mathbf{u}\in C[t_{0},\infty]\) is a control for the functional Equation (19), subject to state Equation (3), and if \(\mathbf{x}\) denotes the corresponding state, then there exists a \(\mathbf{p}\in C[t_{0},\infty]\) such that  
\[\frac{\partial H}{\partial\mathbf{u}}(\mathbf{p},\mathbf{x},\mathbf{u},t)= \mathbf{0}\quad for\quad t\in[t_{0},\infty] \tag{6}\]  
and  
\[\mathbf{\dot{p}}=-\frac{\partial H}{\partial\mathbf{x}}(\mathbf{p},\mathbf{x },\mathbf{u},t)\quad for\quad t\in[t_{0},\infty]\quad and\quad\mathbf{p}( \infty)=\mathbf{0}. \tag{7}\]Equations (6) and (7) are conditions allowing us to obtain suboptimal control while minimizing Equation (19).