# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 1 Introduction  
Over the last few years, an increasing research effort has been invested in the analysis, design and implementation of distributed damping systems (DDSs) for seismic protection of buildings and civil structures [1; 2]. That kind of passive energy-dissipation systems is formed by a set of damping elements installed at suitable locations of the structure. DDSs are simple, reliable and robust and, when properly designed, are able to produce a remarkable reduction of the overall seismic vibrational response [3; 4; 5]. Broadly speaking, three main issues related to the damping elements have to be addressed in DDS design--(i)_technical setup determination_, (ii) _allocation_ and (iii) _tuning_. Determination of the technical setup is a preliminary step, which is strongly conditioned by the particular problem under consideration and involves selecting a particular kind of damping devices and, possibly, setting other technical characteristics such as the type of damper mounting, maximum damping capacity or maximum stroke [6; 7]. It can also include some overall characteristics as the total number of allowed dampers or the overall damping capacity. The allocation problem requires determining a suitable set of structural positions to implement the dampers, and the tuning problem consists in computing adequate values for the damper parameters. For a given technical setup, the allocation and tuning issues are clearly interlinked and both must be simultaneously solved in order to obtain a DDS design with high-performance characteristics. The combined allocation-tuning problem can be formulated as a constrained optimization problem, with a set of decision variables that describes the different allocation schemes and parameter values of the damping devices, an objective function that allows evaluating the suitability of the corresponding DDSs, and a system of constraints that incorporate relevant features of the considered technical setup. To solve allocation-tuning optimization problems (ATOPs), a wide variety of computational strategies have been proposed, which include specialized optimization methods [8; 9; 10; 11; 12] and adaptations of general-purpose optimization algorithms [13; 14; 15]. Specialized optimization methods are specifically designed for particular kinds of ATOPs and, sometimes, can produce remarkably effective results for the considered problem. However, it should be highlighted that solving ATOPs for large structural systems can be a hard computational task, due to a number of factors such as high dimensionality, the combination of discrete and continuous decision variables, the presence of complex structural constraints, and the computational cost associated to the evaluation of the objective function. In this context, taking advantage of state-of-the-art general-purpose optimization solvers can be an element of critical relevance. The general goal of the paper is to design high-performance DDSs for seismic protection of multibuilding systems (MBSs) formed by a row of \(m\) adjacent buildings as the one schematically displayed in Figure 1. There certainly exists a large number of works on DDS design for single buildings in the literature; some recent discussions on the topic can be found in References [5; 6; 7; 15; 16]. For the particular case of \(m=2\) adjacent buildings, the number of references is remarkably smaller but yet significant. Works in this line include different kinds of interbuilding damping devices [17; 18; 19] and actuation schemes [20; 21; 22]. In contrast, to our best knowledge, the more general case of DDS design for MBSs with \(m\geq 3\) adjacent buildings remains practically unexplored. To keep the complexity of the considered problem within a reasonable level, in this work we assume that all the buildings have identical dynamic characteristics [20; 21]. That choice allows simplifying the notations in the overall mathematical model and can help to clarify the effects produced by the DDS, which otherwise could be confounded by the action of the distinct building responses.