## 3 Control of Underactuated Systems with Variant Constraints
Then, systems (21), (22), (23) or (24) can be globally exponentially stabilized to \(\mathbf{x}^{d}(t)\) by the switched controllers \(\mathbf{v}_{\sigma(\Delta t_{i})}(t)\) according to the switching sequence \((\sigma(\Delta t_{1}),\sigma(\Delta t_{2}),\ldots,\sigma(\Delta t_{N_{\sigma} }))\). For the linear systems (30), the corresponding error systems can be rewritten as  
\[\dot{\mathbf{e}}_{\sigma}=\mathbf{A}_{\sigma}\mathbf{e}_{\sigma}+\tilde{ \mathbf{v}}_{\sigma}, \tag{31}\]  
where \(\mathbf{e}_{\sigma}=\mathbf{x}_{\sigma}(t)-\mathbf{x}_{\sigma}^{d}(t)\) and \(\tilde{\mathbf{v}}_{\sigma}=\mathbf{A}_{\sigma}\mathbf{x}_{\sigma}^{d}-\dot {\mathbf{x}}_{\sigma}^{d}+\mathbf{B}_{\sigma}\mathbf{v}_{\sigma}\). Then, by the feedback \(\tilde{\mathbf{v}}_{\sigma}=-\mathbf{K}_{\sigma}\mathbf{e}_{\sigma}\), the error systems (31) can be written as  
\[\dot{\mathbf{e}}_{\sigma}=\tilde{\mathbf{A}}_{\sigma}\mathbf{e}_{\sigma}, \tag{32}\]  
where the matrices \(\tilde{\mathbf{A}}_{\sigma}=\mathbf{A}_{\sigma}-\mathbf{K}_{\sigma}\) are Hurwitz matrices. Thus, for all \(t\in\Delta t_{i\in S}\), the error systems of the closed-loop subsystems satisfy  
\[\|\mathbf{e}_{\sigma}(t)\|_{t\in\Delta t_{i\in S}}\leq e^{-\rho_{i}t}\left\| \mathbf{e}_{\sigma}(t_{i-1})\right\|, \tag{33}\]  
where \(\rho_{i}>0\) are positive constants that are determined by the characteristic values of the matrices \(\tilde{\mathbf{A}}_{\sigma}\); meanwhile, according to (A4) and (A5), the errors of the open loop subsystems satisfy  
\[\|\mathbf{e}_{\sigma}(t)\|_{t\in\Delta t_{i\in S}}\leq\|\mathbf{e}_{\sigma}(t _{i-1})\|+\delta_{i}, \tag{34}\]  
where \(\delta_{i}>0\) are constants that are caused by the state jumps of the hybrid systems (21), (22), (23) or (24), and the constants \(\delta_{i}\) are finite since the matrix \(\mathbf{M}_{\mathrm{diag}}(\mathbf{q}_{s})\) is always positive definite and the norm \(\|\mathbf{W}(\mathbf{q})\|\) has an upper bound for all smooth constraints Equation (2). On the other hand, due to \(\Omega\
eq\varnothing\) and \(S-\Omega\
eq\varnothing\), on a subsequent interval \(\Delta t_{j\in S,j\
eq i}\), the state errors \(\delta_{i}\) caused by the \(i\)-th state jump can be eliminated because of the assumption (A6), which supposes that \(\tau^{*}\) is sufficiently large. Then, for the given target trajectory \(\mathbf{x}^{d}(t)\) on \(t\in[0,a]\), we can conclude that there exist constants \(0<\bar{\rho}_{i}<\rho_{i}\), \(i\in S\) so that \[ \begin{array}{l}\left\|\mathbf{e}(t_{i})\right\|_{i=N_{\
u}}\leq\left\|\mathbf{e} _{x}(t_{i})\right\|_{i=N_{\
u}}+\left\|\mathbf{e}_{s}(t_{i})\right\|_{i=N_{ \
u}}\\ \qquad\leq e^{-\bar{\rho}_{N_{\
u}}\tau^{*}}\left(\left\|\mathbf{e}_{x}(t) \right\|_{i=N_{\
u}-1}+\left\|\mathbf{e}_{s}(t)\right\|_{i=N_{\
u}-1}\right)\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\cdots\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq e^{- \left(\sum\limits_{i=1}^{N_{\
u}}\bar{\rho}_{i}\right)\tau^{*}}\left(\left\| \mathbf{e}_{x}(t)\right\|_{i=1}+\left\|\mathbf{e}_{s}(t)\right\|_{i=1}\right) \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq e^{-\bar{\rho}_{0}\tau^{*} }\left(\left\|\mathbf{e}_{x}(0)\right\|+\left\|\mathbf{e}_{s}(0)\right\| \right).\end{array}  \tag{35}\]  
This completes the proof. **Remark 4**: _Theorem 3 reveals that a class of the underactuated systems with variant constraints can track certain time varying trajectory by switched linear controllers with only partial state feedback in a time-sharing manner, even though the underactuated systems under consideration are generally nonholonomic systems that are possibly caused by either external non-integrable differential constraints or internal dynamics of the passive subsystems (28) and (29). Thus, the presented switched linear controllers are essentially a class of discontinuous nonlinear feedback controllers [27].