# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 3 Optimization Procedure
### Allowed Damper Positions, Dampers Allocation and Damping Coefficients  
In order to determine the structure of the interstory damping systems, we introduce the scheme of allowed damper positions \(\hat{\sigma}=[\hat{\sigma}^{(1)},\ldots,\hat{\sigma}^{(m)}]\), where \(\hat{\sigma}^{(j)}=[\hat{\sigma}^{j}_{1},\ldots,\hat{\sigma}^{j}_{\hat{\eta}_ {j}}]\) is a list that indicates the story levels in building \(\mathcal{B}^{(j)}\) at which additional interstory dampers can be implemented and \(\hat{\eta}_{j}\) is the number of such allowed positions. When no additional interstory dampers are allowed in building \(\mathcal{B}^{(j)}\), we agree that \(\hat{\sigma}^{(j)}\) is an empty list with \(\hat{\eta}_{j}=0\) elements. Analogously, to specify the structure of the interbuilding damping system, we introduce the scheme of allowed positions \(\tilde{\sigma}=[\tilde{\sigma}^{(1)},\ldots,\tilde{\sigma}^{(m-1)}]\), where \(\hat{\sigma}^{(j)}=[\hat{\sigma}^{j}_{1},\ldots,\hat{\sigma}^{j}_{\hat{\eta}_ {j}}]\) is a list that indicates the story levels between buildings \(\mathcal{B}^{(j)}\) and \(\mathcal{B}^{(j+1)}\) at which additional interbuilding dampers can be implemented and \(\tilde{\eta}_{j}\) is the number of those allowed positions. Also in this case, the value \(\hat{\eta}_{j}=0\) indicates that \(\hat{\sigma}^{(j)}\) is an empty list and that no interbuilding dampers can be implemented between \(\mathcal{B}^{(j)}\) and \(\mathcal{B}^{(j+1)}\). The overall scheme of allowed damper positions  
\[\overline{\sigma}=[\hat{\sigma}^{(1)},\ldots,\hat{\sigma}^{(m)},\tilde{\sigma }^{(1)},\ldots,\tilde{\sigma}^{(m-1)}] \tag{44}\]  
has \(\overline{\eta}=\hat{\eta}+\tilde{\eta}\) elements with \(\hat{\eta}=\sum_{j=1}^{m}\hat{\eta}_{j}\) and \(\tilde{\eta}=\sum_{j=1}^{m-1}\tilde{\eta}_{j}\). To illustrate the introduced definitions, let us consider the scheme of allowed damper positions for the three-building system displayed in Figure 4, which permits implementing interstory dampers at the story levels 1-3 in buildings \(\mathcal{B}^{(1)}\) and \(\mathcal{B}^{(3)}\) (blue dashed squares) and interbuilding dampers at the interbuilding levels 4 and 5 (red dotted squares). In this case, we have  
\[ \begin{array}{llllll}\hat{\sigma}^{(1)}=[1,2,3]&\hat{\sigma}^{(2)}=[\ ]&\hat{\sigma}^{(3)}=[1,2,3]&\tilde{\sigma}^{(1)}=[4,5]&\tilde{\sigma}^{(2)}=[ 4,5]\\ \hat{\eta}_{1}=3&\hat{\eta}_{2}=0&\hat{\eta}_{3}=3&\hat{\eta}_{1}=2&\hat{\eta }_{2}=2\end{array}  \tag{45}\]  
with a total number of allowed damper positions \(\overline{\eta}=10\). To define the damper placement positions of an admissible interstory damping system for building \(\mathcal{B}^{(j)}\), we introduce the allocation list \(\hat{\alpha}^{(j)}=[\hat{\alpha}^{j}_{1},\ldots,\hat{\alpha}^{j}_{\hat{\eta}_ {j}}]\), where \(\hat{\alpha}^{j}_{i}\) is a binary variable that takes the values  
\[\hat{\alpha}^{j}_{i}= \begin{cases}1&\text{if an interstory damper is implemented at position $\hat{\alpha}^{j}_{i}$ in $\mathcal{B}^{(j)}$,}\\ 0&\text{otherwise.}\end{cases}  \tag{46}\]The number of interstory dampers implemented in \(\mathcal{B}^{(j)}\) is \(\hat{n}_{j}=\sum_{i=1}^{\hat{\eta}_{j}}\hat{\alpha}_{i}^{j}\) and \(\hat{n}=\sum_{j=1}^{m}\hat{n}_{j}\) is the total number of interstory dampers. Analogously, the damper placement positions for an admissible interbuilding damping system between buildings \(\mathcal{B}^{(j)}\) and \(\mathcal{B}^{(j+1)}\) are defined by the allocation list of binary variables \(\tilde{\alpha}^{(j)}=[\tilde{\alpha}_{1}^{j},\ldots,\tilde{\alpha}_{\hat{ \eta}_{j}}^{j}]\), where \(\tilde{\alpha}_{i}^{j}=1\) indicates that an interbuilding damper is implemented at level \(\tilde{\sigma}_{i}^{j}\). The number of dampers implemented between \(\mathcal{B}^{(j)}\) and \(\mathcal{B}^{(j+1)}\) is \(\tilde{n}_{j}=\sum_{i=1}^{\hat{\eta}_{j}}\tilde{\alpha}_{i}^{j}\) and \(\tilde{n}=\sum_{j=1}^{m-1}\tilde{n}_{j}\) is the total number of interbuilding dampers.