Introduction

Abstract

The aim of this work is to study the asymptotic behavior of various classes of optimal control problems governed by partial differential equations (PDEs) on domains with a lattice-type structure. The investigation of optimal control problems for such structures is important to researchers working with cellular materials (lightweight materials) such as honeycomb structures, ceramic, metal, or polymeric foams which are used in automotive industry, aircraft design, robotics, and micro-mechanics. Lattice-type materials are also important in life sciences. In particular, bones and, correspondingly, bio-mimetic materials are subject of major research initiatives. Other modern engineering applications are flexible multistructures and civil engineering technologies for transportation. Because of the complicated geometry of lattice-type structures, a direct numerical simulations of solutions, let alone of optimal controls, in such a domain is extremely demanding due to the excessive number of variables. Consequently, model reduction is in order. One way of thinking about model reduction is to consider geometrically motivated or material-related scale parameters that tend to zero in order to achieve a new so-called effective model. In the asymptotic limit, the problem may be lower dimensional, as for the 2D-plate models generated via asymptotic analysis out of 3D elasticity, or they can be higher dimensional, as in the case of thin graphlike structures when the spacing scale tends to zero. Regardless of the dimension of the limiting system versus the dimension of the original model, the limiting problem should be easier to access numerically. In this monograph, asymptotic analysis is the main approach to study optimal control of boundary value problems in such complex domains because it gives not only the possibility to replace the original problems by the corresponding limit problems defined in “simpler” domains but also to use an optimal solution to the limit problem as the basis for the construction of suboptimal controls for the original control problem.