## 3 Modified SDRE Method  
Let Equation (18) be the continuous time nonlinear system  
\[\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})+\mathbf{B}\mathbf{u}, \tag{18}\]where \(\mathbf{x}\in\mathbb{R}^{n}\) and \(\mathbf{u}\in\mathbb{R}^{m}\) are the state and input vectors. \(\mathbf{F}(\mathbf{x})\) 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 \(t_{0}\leq t\) that minimizes the performance index [28; 29; 30; 31]  
\[J(\mathbf{u})=\frac{1}{2}\int\limits_{0}^{\infty}(\mathbf{x}^{T}\mathbf{Q} \mathbf{x}+\mathbf{u}^{T}\mathbf{R}\mathbf{u})dt, \tag{19}\]  
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. The first modification (compared to the classic approach) is to rewrite the system Equation (18) as a sum of state-independent and nonlinear state-dependent coefficient (SDC) forms Equation [25]:  
\[\dot{\mathbf{x}}=\mathbf{A}(\mathbf{x})\mathbf{x}+\mathbf{B}\mathbf{u}= \mathbf{A}_{1}\mathbf{x}+\mathbf{A}_{2}(\mathbf{x})\mathbf{x}+\mathbf{B} \mathbf{u}, \tag{20}\]  
where  
\[\mathbf{A}(\mathbf{x})=\mathbf{A}_{1}+\mathbf{A}_{2}(\mathbf{x}) \tag{21}\]  
is a sum of a constant matrix and a state-dependent matrix [12; 33]. The SDC parameterization of Equation (20) can be performed the same way as for Equation (4), for all \(0\leq\alpha\leq 1\)[32]:  
\[\alpha[\mathbf{A}_{1,I}+(1-\alpha)\mathbf{A}_{1,II}+\alpha\mathbf{ A}_{2,II}(\mathbf{x})]\mathbf{x}=\] \[\alpha[\mathbf{A}_{1,I}+(1-\alpha)\mathbf{A}_{1,II}+\alpha \mathbf{A}_{2,I}(\mathbf{x})+(1-\alpha)\mathbf{A}_{2,II}(\mathbf{x})]\mathbf{ x}= \tag{22}\] \[\alpha\mathbf{F}_{I}(\mathbf{x})+(1-\alpha)\mathbf{F}_{II}( \mathbf{x})=\mathbf{F}(\mathbf{x}).\]  
To easily formulate controllability criteria, the state-dependent controllability matrix is introduced \(\mathbf{W}(\mathbf{x})\)[25]  
\[\mathbf{W}(\mathbf{x})=[\mathbf{B}\quad(\mathbf{A}_{1}+\mathbf{A}_{2}( \mathbf{x}))\mathbf{B}\quad\ldots\quad(\mathbf{A}_{1}+\mathbf{A}_{2}(\mathbf{ x}))^{n-1}\mathbf{B}]. \tag{23}\]  
If \(\mathbf{W}(\mathbf{x})\) (state-dependent) has the full rank, then the system is controllable for all \(\mathbf{x}\in\mathbb{R}^{n}\)[34]. As in the previous case of the SDRE method, an admissible control \(\mathbf{u}(t)\in\mathbb{R}^{m}\), that minimizes the performance index can be found for [28; 29; 30; 31]  
\[J(\mathbf{u})=\frac{1}{2}\int\limits_{0}^{\infty}(\mathbf{x}^{T}\mathbf{Q} \mathbf{x}+\mathbf{u}^{T}\mathbf{R}\mathbf{u})dt, \tag{24}\]  
where \(\mathbf{Q}\in\mathbb{R}^{n\times n}\) is a symmetric positive semi- definite matrix, and \(\mathbf{R}\in\mathbb{R}^{m\times m}\) is a symmetric positive definite matrix. The control problem for the nonlinear continuous-time system Equation (18) 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 controls the system state vector from \(\mathbf{x}_{0}\) to \(\mathbf{x}_{\infty}\) while minimizing the performance index (Equation (19)). If the integrand of the cost 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 \(J(\mathbf{u}):C[t_{0},\infty]\rightarrow\mathbb{R}\). Note that, in this approach, two feedback compensators are defined \(\mathbf{A}(\mathbf{x})=\mathbf{A}_{1}+\mathbf{A}_{2}(\mathbf{x})\).