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.
