## 4 Minimizing Input
### Definition and Properties  
To define the minimizing input formally, we first prove the uniqueness of the optimal solution in the case of \(L_{g}V(t,e)\
eq 0\):  
**Proposition 1**. : _Consider the convex optimization problem (36) for any fixed \((t,e)\in[0,\infty)\times\mathbb{R}^{n}\) such that \(L_{g}V(t,e)\
eq 0\). Then, \(\bar{u}_{t,e}\in\tilde{U}(t,e)\) satisfying the KKT condition (40), (41) is unique. In other words, \(\Phi(t,e)=\{\bar{u}_{t,e}\}\) holds._  
To prove Proposition 1, we employ the following lemma:  
**Lemma 5**. : _Consider the optimization problem (36) for any fixed \((t,e)\) such that \(L_{g}V(t,e)\
eq 0\). Then, the following holds:_  
\[\Phi(t,e)\subset\partial\tilde{U}(t,e). \tag{48}\]  
Proof. : Let \(\bar{u}_{t,e}\in\Phi(t,e)\). We suppose that \(\bar{u}_{t,e}\in\tilde{\tilde{U}}(t,e)\). Then, there exists a constant \(\delta>0\) such that  
\[u^{\prime}:=\bar{u}_{t,e}-\delta L_{g}V^{\top}(t,e)\in\tilde{U}(t,e). \tag{49}\]  
Since \(\bar{u}_{t,e}\) is an optimal solution,  
\[\min_{\bar{u}\in\tilde{U}(t,e)}L_{g}V(t,e)\cdot\bar{u}=L_{g}V(t,e)\cdot\bar{u} _{t,e} \tag{50}\]  
holds. By considering \(u^{\prime}\), however,  
\[ \begin{split} L_{\tilde{g}}V(t,e)\cdot u^{\prime}& =L_{\tilde{g}}V(t,e)\cdot[\bar{u}_{t,e}-\delta L_{\tilde{g}}V(t,e)] \\ &=L_{g}V(t,e)\cdot\bar{u}_{t,e}-\delta\|L_{g}V(t,e)\|^{2}<L_{g}V \cdot\bar{u}_{t,e}.\end{split}  \tag{51}\]  
This contradicts the fact that \(\bar{u}_{t,e}\in\Phi(t,e)\). Proof of Proposition 1.: We suppose that \(\bar{u}_{1},\bar{u}_{2}\in\Phi(t,e)\) and \(\bar{u}_{1}\
eq\bar{u}_{2}\). Let \(\beta\in(0,1)\) be a constant. Then, for \(u^{\prime}=\beta\bar{u}_{1}+(1-\beta)\bar{u}_{2}\in\tilde{U}(t,e)\), we can obtain  
\[ \begin{split} L_{g}V(t,e)\cdot u^{\prime}&=L_{g}V( t,e)\cdot[\beta\bar{u}_{1}+(1-\beta)\bar{u}_{2}]\\ &=\beta L_{g}V(t,e)\cdot\bar{u}_{1}+(1-\beta)L_{g}V(t,e)\cdot \bar{u}_{2}=\min_{\bar{u}\in\tilde{U}(t,e)}L_{g}V(t,e)\cdot\bar{u}.\end{split}  \tag{52}\]  
This shows that \(u^{\prime}\) is also an optimal solution. Since the set \(\tilde{U}(t,e)\subset\mathbb{R}^{m}\) is strictly convex by (H1') of Lemma 1, however,  
\[u^{\prime}=\beta\bar{u}_{1}+(1-\beta)\bar{u}_{2}\in\tilde{\tilde{U}}(t,e). \tag{53}\]  
This contradicts Lemma 5. Thanks to Proposition 1, we can formally define the minimizing input as follows:  
**Definition 3** ( Minimizing input ). : _The following state feedback \(\bar{k}:[0,\infty)\times\mathbb{R}^{n}\to\mathbb{R}^{m}\) is said to be the minimizing input for the error system (3) with respect to a given TCLF \(V(t,e)\):_  
\[\bar{k}(t,e)= \begin{cases}\operatorname*{argmin}_{a\in\tilde{U}(t,e)}L_{g}V \cdot\bar{u}&(L_{g}V(t,e)\
eq 0)\\ 0&(L_{g}V(t,e)=0)\end{cases}  \tag{54}\]  
**Remark 8**. : _When \(L_{g}V(t,e)=0\), any \(\tilde{u}\in\tilde{U}(t,e)\) becomes an optimal solution (see Remark 6). The reason we set \(\tilde{k}(t,e)=0\) in Definition 3 is to design a trajectory-tracking controller that is continuous even when \(L_{g}V(t,e)=0\) in Section 5._  
In the following, we introduce two important properties of \(\tilde{k}(t,e)\) on the set  
\[\mathcal{X}_{1}:=\{(t,e)\ |\ L_{g}V(t,e)\
eq 0\}\subset[0,\infty)\times\mathbb{R }^{n}. \tag{55}\]  
**Lemma 6**. : _The minimizing input (54) satisfies_  
\[L_{\tilde{g}}V(t,e)\cdot\tilde{k}(t,e)<0,\ \forall(t,e)\in\mathcal{X}_{1}. \tag{56}\]  
Proof. : Fix \((t,e)\in\mathcal{X}_{1}\) arbitrarily. According to Lemma 1, there exists a constant \(\delta>0\) such that  
\[\tilde{B}^{m}(0,\delta)\subset\tilde{U}(t,e). \tag{57}\]  
Thanks to Lemma 4, we have  
\[ \begin{split} L_{\tilde{g}}V(t,e)\cdot\tilde{k}(t,e)& =\min_{a\in\tilde{U}(t,e)}L_{\tilde{g}}V(t,e)\cdot\tilde{u}\\ &\leq\min_{\tilde{u}\in\tilde{B}^{m}(0,\delta)}L_{\tilde{g}}V(t,e )\cdot\tilde{u}=L_{\tilde{g}}V(t,e)\cdot\left(-\frac{\delta}{\|L_{\tilde{g}}V (t,e)\|}\right)L_{\tilde{g}}V^{\top}(t,e)\\ &=-\delta\|L_{\tilde{g}}V(t,e)\|<0.\end{split}  \tag{58}\]  
**Lemma 7** (Continuity of \(\tilde{k}(t,e)\)). : _The minimizing input (54) is continuous on \(\mathcal{X}_{1}\)._  
Proof.