## 2 Problem Statement  
Let us consider a continuous-time LPV system  
\[\dot{x}(t) = A(\theta(t))x(t)+B(\theta(t))u(t),\] \[y(t) = C(\theta(t))x(t)+D(\theta(t))u(t),\]where \(x(t)\in\mathbb{R}^{n}\) is the state vector, \(u(t)\in\mathbb{R}^{m}\) is the input vector and \(\theta(t)\in\Theta\subset\mathbb{R}^{n_{\theta}}\) is the scheduling parameter vector, with \(\Theta\) known, closed and bounded set. Matrices \(A(\theta(t))\), \(B(\theta(t))\), \(C(\theta(t))\) and \(D(\theta(t))\) are the parameter-dependent state, input, output and feedforward matrices, respectively. The polytopic representation of (1) is used throughout this paper. In this representation, the system's matrices are defined as a weighted sum of matrices that represent the system in the \(N\) vertices of a polytope that contains \(\Theta\)  
\[ \begin{split}\dot{x}(t)&=\sum_{i=1}^{N}\mu_{i}( \theta(t))(A_{i}x(t)+B_{i}u(t)),\\ y(t)&=\sum_{i=1}^{N}\mu_{i}(\theta(t))(C_{i}x(t)+ D_{i}u(t)),\end{split}  \tag{2}\]  
where matrices \(A_{i}\), \(B_{i}\), \(C_{i}\) and \(D_{i}\) define the so-called _vertex systems_ and \(\mu_{i}\) are the coefficients of the polytopical decomposition that satisfy  
\[\sum_{i=1}^{N}\mu_{i}(\theta(t))=1,\;\mu_{i}(\theta(t))\geq 0,\;\forall i=1, \ldots,N. \tag{3}\]  
The time dependency of \(x\), \(\theta\), \(y\) and \(u\) is dropped from now on and it will only be made explicit when necessary. Also, without loss of generality, we consider the behaviour of the system starting from a time instant \(t_{0}=0\). The extension to the case where \(t_{0}\
eq 0\) is straightforward by means of a simple translation of the time axis. The following assumptions are made on (2):  
**Assumption 1**. : _The state variables and the scheduling variables are measurable or can be estimated online._  
**Assumption 2**. : _The input and output matrices are constant._  
**Assumption 3**. : _System disturbances are not considered._  
**Assumption 4**. : _The system (2) is stabilizable._  
**Remark 1**. : _Note that Assumptions 1-3 are only made for the sake of keeping the mathematical complexity somehow simpler and could be removed by extending the results presented in this paper taking into account existing techniques in the literature. For instance, inexactly measured parameters were considered by Reference [26]; the complexity arising from parameter-varying input and output matrices can be dealt with using conditions based on [PERSON]'s theorems [27]; disturbances can be considered under a quadratic boundedness framework, see for example, Reference [28]. On the other hand, Assumption 4 is a necessary (not sufficient) condition in order to solve the controller design problem described in this paper. Note that recent work has suggested a practical test to assess this property in systems described by a polytopic representation [29]._  
Considering the above assumptions, the output equation can be neglected and (2) becomes  
\[\dot{x}=\sum_{i=1}^{N}\mu_{i}(\theta)A_{i}x+Bu. \tag{4}\]  
In this paper, we consider the case in which the input signal is affected by a nonlinearity, such that the change \(Bu\to Bsat(u)\) arises in (4), with \(sat(u)\) denoting a symmetric saturation  
\[sat(u)= \begin{cases}sign(u)\gamma&\text{if}\;\;\mid u\mid>\gamma\\ u&\text{if}\;\;\mid u\mid\leq\gamma\end{cases} , \tag{5}\]where \(>\) and \(\leq\) are meant element-wise and \(\gamma\in\mathbb{R}_{+}^{m}\) is the saturation limit value, which is considered constant in Section 3 and time-varying within the interval \([\gamma_{j},\overline{\gamma_{j}}]\) in Section 4. The contribution of this work lies in proposing conditions to design an LPV state-feedback controller that ensures the stability of the system (4).