## 4 Stability Analysis  
Asymptotic stability of the closed-loop system ensures the possibility of controlling the states from the initial values to the final ones. The controlled system with the SDRE compensator-based feedback is locally asymptotically stable [25]. Let \(r>0\) be the largest radius, such that \(\mathbb{B}_{r}(0)\subseteq\Omega\). Assuming that the system is stabilizable at \(\mathbf{x}=\mathbf{0}\), it is possible to use optimal control theory to define matrix \(\mathbf{K}_{1}\), such that all eigenvalues of \((\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})\) have negative real parts. There exist \(\beta>0\), such that \(Re(\mathbf{p})<-\beta\) for all eigenvalues \(\mathbf{p}\) of \((\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})\), having the system  
\[\hat{\mathbf{x}}=(\mathbf{A}_{1}-\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T} \mathbf{K}_{1})\mathbf{x}+(\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{R}^{ -1}\mathbf{B}^{T}\mathbf{K}_{2}(\mathbf{x}))\mathbf{x}, \tag{43}\]  
where \((\mathbf{A}_{1}+\mathbf{A}_{2}(\mathbf{x}))\mathbf{x}\) and \(\frac{\partial(\mathbf{A}_{1}\mathbf{x}+\mathbf{A}_{2}(\mathbf{x})\mathbf{x})} {\partial\mathbf{x}}\) are continuous in \(\mathbf{x}\) for \(||\mathbf{x}||<r\), where \(r>0\) is the largest radius around the origin \(\mathbf{x}=\mathbf{0}\). Let  
\[\mathbf{g}(\mathbf{x})=\mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}( \mathbf{x}), \tag{44}\]  
and \(\mathbf{h}(\mathbf{x})=\mathbf{g}(\mathbf{x})\mathbf{x}\); then, the system (34) is described by  
\[\dot{\mathbf{x}}=(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})\mathbf{x}+\mathbf{h }(\mathbf{x}). \tag{45}\]  
Evaluation of \(\mathbf{h}(\mathbf{x})\) shows that it is an almost linear system:  
\[\lim_{||\mathbf{x}||\to 0}\frac{||\mathbf{h}(\mathbf{x})||}{||\mathbf{x}||}=0, \tag{46}\]  
from the inequality  
\[||\mathbf{h}(\mathbf{x})||\leq||\mathbf{g}(\mathbf{x})||||\mathbf{x}||=|| \mathbf{A}_{2}(\mathbf{x})-\mathbf{B}\mathbf{K}_{2}(\mathbf{x})||||\mathbf{x}|. \tag{47}\]  
Thus, \(||\mathbf{g}(\mathbf{x})||\to 0\) when \(||\mathbf{x}||\to 0\); then, \(\mathbf{h}(\mathbf{x})\) satisfies the condition Equation (46). The theoretical considerations of almost linear systems leads to the conclusion that, if eigenvalues \((\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})\) have negative real parts, \(\mathbf{h}(\mathbf{x})\) is continuous around the origin, and the condition of Equation (46) holds, then \(\mathbf{x}=\mathbf{0}\) is asymptotically stable [38]. Let \(\delta>0\); then, there exists a \(\hat{\eta}\in(0,r)\), such that \(||\mathbf{h}(\mathbf{x})||\leq\delta||\mathbf{x}||\) whenever \(||\mathbf{x}||\leq\hat{\eta}\). Let \(\mathbf{x}(t_{0})=\mathbf{x}_{0}\in\mathbb{B}_{\hat{\eta}}(0)\); assuming that \(f\) is continuous, the solution exists and remains in \(\mathbb{B}_{\hat{\eta}}(0)\) for \(t\in[t_{0},\infty]\), then  
\[\mathbf{x}(t)=e^{(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})t}\mathbf{x}(t_{0}) +\int_{t_{0}}^{t}e^{(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})(t-s)}\mathbf{h}( \mathbf{x}(s))ds. \tag{48}\]  
Considering the norm and the assumption \(||\mathbf{x}(t)||<\hat{\eta}\), it is possible to obtain  
\[||\mathbf{x}(t)||\leq\left||e^{(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})t} \right||||\mathbf{x}(t_{0})||+\delta\int_{t_{0}}^{t}\Bigl{|}\Bigl{|}e^{( \mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})(t-s)}\Bigr{|}\Bigr{|}||\mathbf{x}(s) ||ds. \tag{49}\]  
There exists a positive constant \(G\) and \(\beta\), such that  
\[\Bigl{|}\Bigl{|}e^{(\mathbf{A}_{1}-\mathbf{B}\mathbf{K}_{1})t}\Bigr{|}\Bigr{|} \leq Ge^{-\beta t} \tag{50}\]  
and  
\[||\mathbf{x}(t)||\leq Ge^{-\beta t}||\mathbf{x}(t_{0})||+\delta G\int_{t_{0}} ^{t}e^{-\beta(t-s)}||\mathbf{x}(s)||ds. \tag{51}\]  
Referring to the Gronwall inequality and after multiplication by \(e^{\beta t}\), we obtain  
\[||\mathbf{x}(t)||\leq G||\mathbf{x}(t_{0})||e^{-(\beta-\delta\mathbf{G})t} \tag{52}\]  
for all \(t>0\), such that \(||\mathbf{x}(t)||<\hat{\eta}\).