## 3 Stabilization Problem
### Main Result
However, the converse is not true: a matrix \(P\) satisfying (25) might not satisfy (28). Moreover, a solution (25) may not be a solution of (27) and conversely a solution of (27) may not be a solution of (25). An interesting question is: Are there systems described by (1)which do not satisfy conditions of Lemma 1, but satisfy Assumption A2? It is not easy to answer such a question because checking whether Assumption A2 is satisfied is a difficult task due to its dependence with respect to the state of the system. We introduce now a key lemma, important for deriving the main result of this section. **Lemma 2**. : _Consider \(x_{e}\in X_{eq}\) and matrix \(P\) solution of (25). Suppose that Assumption A2 is verified. Then, we have_  
\[\min_{\lambda\in\Lambda_{S}}e^{T}(t)P[A_{0}e(t)+B_{0}(x(t))\lambda+A_{0}x_{e}] <-e(t)^{T}Q_{1}e(t) \tag{31}\]  
Proof. : We have by convexity  
\[ \begin{array}{rl}\min_{\lambda\in\Lambda_{S}}e^{T}(t)P[A_{0}e(t)+B_{0}(x(t) )\lambda+A_{0}x_{e}]&=\min_{\lambda\in\Lambda_{S}}e^{T}(t)P[A_{0}e(t)+B_{0}(x (t))\lambda+A_{0}x_{e}]\\ &=e(t)^{T}PA_{0}e(t)+\min_{\lambda\in\Lambda_{S}}(e)^{T}P[B_{0}(x(t))\lambda+ A_{0}x_{e}]\\ &=e^{T}(t)PA_{0}e(t)+\min_{\lambda\in\Lambda_{S}}e(t)^{T}P[B_{0}(x)\lambda-B_{ 0}(x_{e})\lambda_{e}]\\ &<-e(t)^{T}Q_{1}e(t)\text{ by assumption A2}\end{array}  \tag{32}\]  
The main result of this section can now be stated. Its simple proof results from the above lemma. **Theorem 1**. : _Consider the system (12), \(x_{e}\in X_{eq}\) and matrix \(P\) solution of (25). Suppose that Assumptions A1 and A2 are satisfied. The control \(\lambda(t)\) defined by_  
\[\lambda(t)=\arg\biggl{\{}\min_{d\in\Lambda_{S}}e^{T}(t)P[A_{0}e(t)+B_{0}(x(t) )d+A_{0}x_{e}]\biggr{\}} \tag{33}\]  
_solves Problem 1. In addition, the error signal satisfies_  
\[J(e)=\int_{0}^{\infty}e^{T}(t)Q_{1}e(t)dt\leq\frac{1}{2}e^{T}(0)Pe(0) \tag{34}\]  
Proof. : Consider the Lyapunov function \(V(e(t))=\frac{1}{2}e^{T}(t)Pe(t)\). Then,  
\[\frac{dV(e(t))}{dt} =e(t)^{T}P\frac{de(t)}{dt}=e(t)^{T}P(A_{0}e(t)+B_{0}(x(t))\lambda( t)+A_{0}x_{e}) \tag{35a}\] \[=\min_{d\in\Lambda_{S}}\Bigl{[}e(t)^{T}P(A_{0}e(t)+B_{0}(x(t))d+A _{0}x_{e}\Bigr{]}\] (35b) \[<-e(t)^{T}Q_{1}e(t)\text{ by Lemma 2}. \tag{35c}\]  
Since \(\frac{dV(e)}{dt}<0\) for all \(e\
eq 0\), we conclude that \(\lim_{t\to\infty}e(t)=0\). Now, integrating (35c) from zero to infinity and taking into account that \(V(e(\infty))=0\), we obtain (34). **Remark 3**. : - _When \(m=1\), the control proposed in the previous theorem can be formulated in a simple way. This case is particularly important because it corresponds to the important class of converters controlled by one variable. Several converters belong to this class. The matrix \(B_{0}(x(t))\) and the control \(\lambda(t)\) are then given by_  
\[B_{0}(x(t))=\left[ \begin{array}{cc}B_{1}(x(t))&B_{2}(x(t))\\ \end{array} \right],\ \ \lambda(t)=\left[ \begin{array}{cc}\lambda_{1}(t)& \lambda_{2}(t)\\ \end{array} \right]^{T} \tag{36}\]  
_Theorem 1 leads to the following corollary whose simple proof is omitted._  
**Corollary 1**. : _Consider the system (1) with \(m=1\) and suppose that Assumptions A1 and A2 are satisfied. Then, the control_  
\[\left\{ \begin{array}{ll}\mbox{If }e^{T}(t)P[(A_{2}-A_{1})x(t)+(B_{2}-B_{1})V_{in}] >0\mbox{ then }\lambda(t)=\left[\begin{array}{c}1\\ 0\end{array}\right]\mbox{or }u(t)=0\\ \\ \mbox{If }e^{T}(t)P[(A_{2}-A_{1})x(t)+(B_{2}-B_{1})V_{in}]\leq 0\mbox{ then }\lambda(t)=\left[\begin{array}{c}0\\ 1\end{array}\right]\mbox{or }u(t)=1\end{array} \right. \tag{37}\]  
_solves Problem 1._  
* _The control can also be written as_ \[\lambda(t)=\arg\biggl{\{}\min_{d\in\Lambda_{S}}e^{T}(t)P[A_{0}e(t)+B_{0}(x(t)) d+A_{0}x_{e}]\biggr{\}}=\arg\biggl{\{}\min_{d\in\Lambda_{S}}e^{T}(t)PB_{0}(x(t))d \biggr{\}}\] (38)
* _From inequality (_34_), we see that an over bound of the integral of error depends explicitly on matrix_ \(P\)_. To go in the sense of a minimization of the integral of error, a possible way could be to minimize the trace of_ \(P\) _under the constraint (_25_).