## Two-Building Mathematical Models
### Controlled Configurations  
In order to mitigate the vibrational response of the adjacent buildings, we consider two different kinds of force actuation devices: (i) _interstory actuators_, which are implemented between consecutive stories of the same building, and (ii) _interbuilding actuators_, which are implemented between stories located at the same level in the adjacent buildings. In both cases, the actuation device produces a pair of opposite structural forces on the corresponding stories. An actuation scheme with two interstory actuators (\(d_{1}\) and \(d_{2}\)) located at the buildings' lowest level, and one interbuilding actuator (\(d_{3}\)) implemented at the fourth-story level is schematically depicted in Figure 1b, where \(u_{j}(t)\) denotes the actuation force produced by the actuation device \(d_{j}\). For this control configuration, the lateral motion of the buildings can be described by the second-order differential equation  
\[\mathbf{M}\,\ddot{\mathbf{q}}(t)+\mathbf{C}_{\mathrm{d}}\,\dot{\mathbf{q}}(t) +\mathbf{K}_{\mathrm{s}}\,\mathbf{q}  
\[\mathbf{T}_{\mathrm{u}}=\left[\begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&-1\\ 0&1&0\\ 0&0&0\\ 0&0&0\\ 0&0&1\\ 0&0&0\end{array}\right]. \tag{24}\]  
In this work, we consider a set of six different control configurations, defined by a particular actuation scheme. The configurations that contain interbuilding actuation devices are called _linked_, and those that only contain interstory actuators are called _unlinked_. The _control configuration I_ (CC1), presented in Figure 2a, is an unlinked configuration that includes two interstory actuators (\(d_{1}\) and \(d_{2}\)) located at the buildings' lowest level. The _control configuration II_ (CC2), presented in Figure 2b, is a linked configuration that includes an interstory actuator (\(d_{1}\)) located at the lowest level of the building \(\mathcal{B}^{(2)}\), and an interbuilding actuator (\(d_{2}\)), located at the fourth-story level. Control configurations with three and four actuation devices are presented in Figures 3 and 4, respectively. The control location matrices corresponding to the configurations CC1, CC2 and CC3 are, respectively,  
\[\mathbf{T}_{\mathrm{u}}^{\mathrm{I}}=\left[\begin{array}{ccc}1&0\\ 0&0\\ 0&0\\ 0&0\\ 0&1\\ 0&0\\ 0&0\\ 0&0\end{array}\right],\quad\mathbf{T}_{\mathrm{u}}^{\mathrm{II}}=\left[ \begin{array}{ccc}0&0\\ 0&0\\ 0&0\\ 0&-1\\ 1&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0&0\end{array}\right],\quad\mathbf{T}_{\mathrm{u}}^{\mathrm{III}}=\left[ \begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&0&0\\ 0&1&-1\\ 0&0&1\\ 0&0&0\\ 0&0&0\\ 0&0&0\end{array}\right], \tag{25}\]  
the matrix \(\mathbf{T}_{\mathrm{u}}^{\mathrm{IV}}\) corresponding to the configuration CC4 has been previously presented in Equation (24), and the control location matrices corresponding to the configurations CC5 and CC6 are, respectively,  
\[\mathbf{T}_{\mathrm{u}}^{\mathrm{v}}=\left[\begin{array}{cccc}1&-1&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&1&-1\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right],\quad\mathbf{T}_{\mathrm{u}}^{\mathrm{v}}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&1&-1&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{array}\right]. \tag{26}\]Figure 3: Actuation schemes with three actuation devices. (**a**) Unlinked configuration CC3, with three interstory actuators (\(d_{1}\), \(d_{2}\) and \(d_{3}\)). (**b**) Linked configuration CC4, with two interstory actuators (\(d_{1}\) and \(d_{2}\)) and an interbuilding actuator (\(d_{3}\)).  
Figure 2: Actuation schemes with two actuation devices. (**a**) Unlinked configuration CC1, with two interstory actuators (\(d_{1}\) and \(d_{2}\)). (**b**) Linked configuration CC2, with an interstory actuator (\(d_{1}\)) and an interbuilding actuator (\(d_{2}\)).