## 4 Stability Analysis
\(\delta\) is chosen from \(\delta\in(0,\frac{\beta}{G})\); then, with respect to \(\hat{\eta}\) having \(\gamma\in(0,\hat{\eta})\), it is possible to obtain \(\eta=\)min\(\{\hat{\eta},\frac{\gamma}{G}\}\). For all \(\mathbf{x}_{0}\in\mathbb{B}_{\eta}\), it is possible to obtain \(||\mathbf{x}(t)||<\gamma\leq\hat{\eta}\) for all \(t>0\), and \(\mathbf{x}=\mathbf{0}\) is stable. The condition of Equation (52) is satisfied for all \(t>0\) if \(\mathbf{x}_{0}\in\mathbb{B}_{\eta}\), \(\mathbf{x}=\mathbf{0}\) is asymptotically stable, since \(\beta-\delta G>0\)[25]. For the proposed method, a global uniform stability can also be proved using the Lyapunov function. The proof can be obtained more easily than for classic SDRE, because feedback compensator gains are state-independent, and for a closed-loop system, there is no problem with generalization and definition of an attraction region. Assuming that SDC parameterization is stabilizable and detectable for all \(\mathbf{x}(t)\), then the closed-loop matrix \(\mathbf{A}_{CL}\) is symmetric for all \(\mathbf{x}(t)\), and the solution of SDRE is asymptotically stable when \(t\in[t_{0},\infty]\). Let the control law be in the form  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]0  
and \(\mathbf{K}_{1}\) satisfies ARE  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]1  
while \(\mathbf{K}_{2}(\mathbf{x})\) is taken from  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]2  
Then, the system in closed-loop form  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]3  
is globally asymptotically stable if \(\mathbf{K}_{1}\) is the unique symmetric positive-defined matrix. Let us define the as Lyapunov function  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]4  
and  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]5  
for \(\mathbf{x}\in\mathbb{U}\), where \(\mathbb{U}\) in \(\mathbb{R}^{\pi}\) has the origin, and constants \(k_{1}\) and \(k_{2}\) are obtained from  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]6  
and  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]7  
for \(i=1,2,...,n\). Using Equations (54) and (56), it is possible to obtain  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]8  
Taking Equation (56), assuming that \(\mathbf{K}_{2}(\mathbf{x})\) assures linearization of Equations (45) and (46), and that the solution of ARE, Equation (54), is always symmetric, then  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]9  
Then, taking into consideration (54), it is possible to obtain  
\[\dot{\mathbf{x}}=(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})\mathbf{x}+\mathbf{h }(\mathbf{x}). \tag{45}\]0  
Let us assume that  
\[\dot{\mathbf{x}}=(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})\mathbf{x}+\mathbf{h }(\mathbf{x}). \tag{45}\]1  
where  
\[\dot{\mathbf{x}}=(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})\mathbf{x}+\mathbf{h }(\mathbf{x}). \tag{45}\]2  
for \(\mathbf{x}\in\mathbb{U}\) and \(i=1,2,...,n\). Generally, in an SDRE control problem with infinite time horizon, the existence of constants \(k_{1},k_{2},k_{3}\) do not assure global stability. The closed-loop matrix is state-dependent but can be generalized to global stability by defining the region of attraction [39]. In the proposed method, the existence of constants \(k_{1},k_{2},k_{3}\) proposed in Equations (59), (60), and (65) assures global stability, because the matrix \(\mathbf{K}_{1}\) is state-independent, so there is no reason to define the region of attraction for the Riccati Equation (54).