Optimal Control Problems for Partial Differential Equations on Reticulated Domains Approximation and Asymptotic Analysis

Optimal control of partial differential equations (PDEs) is by now, after more than 50 years of ever increasing scientific interest, a well established discipline in mathematics with many interfaces to science and engineering. During the development of this area, the complexity of the systems to be controlled has also increased significantly, so that today fluid-structure interactions, magneto-hydromechanical, or electromagnetical as well as chemical and civil engineering problems can be dealt with. However, the numerical realization of optimal controls based on optimality conditions, together with the simulation of the states, has become an issue in scientific computing, as the number of variables involved may easily exceed a couple of million.

In order to carry out model-reduction on ever-increasingly complex systems, the authors of this work have developed a method based on asymptotic analysis. They aim at combining techniques of homogenization and approximation in order to cover optimal control problems defined on reticulated domains—networked systems including lattice, honeycomb, and hierarchical structures. The investigation of optimal control problems for such structures is important to researchers working with cellular and hierarchical materials (lightweight materials) such as metallic and ceramic foams as well as bio-morphic material. Other modern engineering applications are chemical and civil engineering technologies, which often involve networked systems. Because of the complicated geometry of these structures—periodic media with holes or inclusions and a very small amount of material along layers or along bars—the asymptotic analysis is even more important, as a direct numerical computation of solutions would be extremely difficult.

Specific topics include:

* A mostly self-contained mathematical theory of PDEs on reticulated domains

* The concept of optimal control problems for PDEs in varying such domains, and hence, in varying Banach-spaces

* Convergence of optimal control problems in variable spaces

* An introduction to the asymptotic analysis of optimal control problems

* Optimal control problems dealing with ill-posed objects on thin periodic structures, thick periodic singular graphs, thick multi-structures with Dirichlet and Neumann boundary controls, and coefficients on reticulated structures

Serving as both a text on abstract optimal control problems and a monograph where specific applications are explored, Optimal Control Problems for Partial Differential Equations on Reticulated Domains is an excellent reference-tool for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains.