## 3 Problem Formulation
: _If the original input constraint is \(\|u\|\leq\alpha\), the transformed constraint for \(\tilde{u}\) becomes_  
\[G_{1}(x,u)=\|u\|-\alpha. \tag{15}\]4  
_Note that (28) is a \"center-shifted\" norm constraint [13] and is not a simple norm constraint for \(\tilde{u}\). This shows that the norm input constraints are not invariant under the input transformation \(\tilde{u}=u-u_{r}(t)\). The important thing here is that the constraint (28) is still convex for \(\tilde{u}\). This is the motivation for introducing the convex input constraints in this article._  
Under Assumption 2, we can prove the following lemma, which shows the invariance of convex input constraints under the input transformation:  
**Lemma 1**. : _Let \(U(x)\) be a convex input constraint, and let Assumption 2 hold. Then, the time-varying convex input constraint \(\tilde{U}(t,e)\) defined by (26) satisfies the following conditions:_  
**(H1')**: _The function_ \(\tilde{G}(t,e,\tilde{u})\) _defined by (_27_) is_ \(C^{1}\) _on_ \([0,\infty)\times\mathbb{R}^{n}\times\mathbb{R}^{m}\) _and is strictly convex with respect to_ \(\tilde{u}\)_;_
**(H2')**: \(\tilde{u}=0\in\tilde{U}(t,e),\ \forall(t,e)\in[0,\infty)\times\mathbb{R}^{n}\)_;_
**(H3')**: _There exists a compact set_ \(\overline{U}^{\prime}\subset\mathbb{R}^{m}\) _such that_ \(\tilde{U}(t,e)\subset\overline{U}^{\prime},\ \forall(t,e)\in[0,\infty)\times \mathbb{R}^{n}.\)__  
Proof. : (H1') \(\tilde{G}\) is clearly \(C^{1}\) since both \(G\) and \(u_{r}\) are \(C^{1}\) with respect to its arguments. Then, we prove the strict convexity of \(\tilde{G}\). For any \(\tilde{u}_{1},\tilde{u}_{2}\in\mathbb{R}^{m}\) and any constant \(\lambda\in(0,1)\), we have  
\[ \begin{split}\tilde{G}(t,e,\lambda\tilde{u}_{1}+(1-\lambda) \tilde{u}_{2})&=G(e+x_{d}(t),u_{r}(t)+\lambda\tilde{u}_{1}+(1- \lambda)\tilde{u}_{2})\\ &=G(e+x_{d}(t),\lambda(u_{r}(t)+\tilde{u}_{1})+(1-\lambda)(u_{r}( t)+\tilde{u}_{2}))\\ &<\lambda G(e+x_{d}(t),u_{r}(t)+\tilde{u}_{1})+(1-\lambda)G(e+x_ {d}(t),u_{r}(t)+\tilde{u}_{2})\\ &=\lambda\tilde{G}(t,e,\tilde{u}_{1})+(1-\lambda)\tilde{G}(t,e, \tilde{u}_{2}),\end{split}  \tag{29}\]  
where the inequality follows from the strict convexity of \(G\). (H2') Owing to Assumption 2, \(G(x,u_{r}(t))<0\) holds for any \((t,x)\). By choosing \(x=e+x_{d}(t)\in\mathbb{R}^{n}\),  
\[G(e+x_{d}(t),u_{r}(t))=\tilde{G}(t,e,u_{r}(t)+0)<0\iff 0\in\tilde{U}(t,e). \tag{30}\]  
(H3') According to (H3) of Definition 2, there exists a constant \(r>0\) such that  
\[U(x)\subset\{u\in\mathbb{R}^{m}\mid\|u\|\leq r\},\ \forall x\in\mathbb{R}^{n}, \tag{31}\]  
and this implies  
\[u\in U(x)\iff\|u\|=\|\tilde{u}+u_{r}(t)\|\leq r. \tag{32}\]By using \(\tilde{u}=(\tilde{u}+u_{r}(t))-u_{r}(t)\) and the triangular inequality, we can obtain  
\[\|\tilde{u}\|\leq\|\tilde{u}+u_{r}(t)\|+\|-u_{r}(t)\|\leq r+\tilde{u}_{r}, \tag{33}\]  
where \(\tilde{u}_{r}=\sup_{t\in[0,\infty)}\|u_{r}(t)\|\). Since \(u\in U(x)\) is equivalent to \(\tilde{u}\in\tilde{U}(t,e)\), we conclude that  
\[\tilde{U}(t,e)\subset\overline{U}^{\prime}:=\{\tilde{u}\in\mathbb{R}^{m}\ |\ \| \tilde{u}\|\leq r+\tilde{u}_{r}\},\ \forall(t,e)\in[0,\infty)\times\mathbb{R}^{n}. \tag{34}\]  
**Remark 4**. : _By Lemma 1, the set \(\tilde{U}(t,e)\subset\mathbb{R}^{m}\) is compact and strictly convex for each fixed \((t,e)\)._  
To consider TCLF-based tracking controller design, we impose the following assumption:  
**Assumption 3**. : _Let \(D\subset\mathbb{R}^{n}\) be a neighborhood of \(e=0\) of the constrained error system (25).