# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 1 Introduction
The ATOP solutions are obtained with  
Figure 1: System of \(m=5\) adjacent buildings equipped with a distributed set of interstory and interbuilding dampers. the genetic algorithm (GA) solver provided by the Matlab Global Optimization Toolbox, which allows using hybrid sets of discrete and continuous optimization variables, permits defining a sufficiently wide variety of optimization constraints and facilitates an easy implementation of parallel computing [31]. To demonstrate the flexibility of the proposed methodology, three different DDSs are designed for the seismic protection of a MBS formed by \(m=5\) adjacent five-story buildings. After that, a proper set of numerical simulations are conducted using the full-scale 180-component of the El Centro 1940 seismic record as ground acceleration disturbance. The obtained results corroborate the effectiveness of the proposed design procedure and confirm its computational efficiency for large-scale problems. The content of the rest of the paper is as follows--in Section 2, a general mathematical model for plain and damped MBSs is presented. The main elements of the optimization procedure are discussed in Section 3. In Section 4, three different DDSs for a five-building system are designed. In Section 5, the corresponding seismic time-responses are computed and compared. Finally, in Section 6, some brief conclusions and future research lines are provided. **Remark 1**: _To reduce the problem complexity and facilitate an effective computational solution, a number of model simplifications have been introduced in the paper, which should be carefully considered for a proper understanding of the scope and applicability of the proposed design methodology. Specifically, buildings are considered as linear multistory planar frames with stories of the same height. In the multibuilding systems, all the buildings are identical, with the same mass, stiffness and damping matrices, and there are no vertical differences between stories placed at the same level in different buildings. The damping elements are ideal linear dampers and the damping constants can take any real value in a prescribed interval. The interbuilding separations are assumed to be large enough to avoid interbuilding impacts. Additionally, the effect of some relevant elements such as the soil-structure interaction and the seismic wave propagation have been neglected._  
**Remark 2**: _The considered problem requires a complex system of notations. To obtain a more organized and clear presentation, throughout this paper symbols related to interstory elements will be usually marked with_ hats _and those corresponding to interbuilding elements will be signaled with_ tides_. When convenient,_ overlines _will be used to distinguish some elements related to the overall MBS. Thus, for example,_ \(\hat{n}_{j}\) _will denote the number of interstory dampers in building_ \(\mathcal{B}^{(j)}\)_, the number of interbuilding dampers between buildings_ \(\mathcal{B}^{(j)}\) _and_ \(\mathcal{B}^{(j+1)}\) _will be indicated by_ \(\bar{n}_{j}\)_, and the overall number of degrees of freedom in the MBS will be represented by_ \(\bar{n}\)_._