Communications in Mathematical Sciences

Volume 11 (2013)

Number 3

Optimal control of a simplified natural convection-radiation model

Pages: 679 – 707

DOI: http://dx.doi.org/10.4310/CMS.2013.v11.n3.a2

Authors

René Pinnau (Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany)

Oliver Tse (Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany)

Abstract

The study of optimal control problems for the two fields of fluid flow and radiation gained considerable attention during the last decade. In this paper we present a comprehensive analysis of an optimal boundary control for a combined natural convection-radiation model, which has applications in the design of combustion chambers or for the control of cooling processes, glass production, or crystal growth. The model under investigation consists of the transient Boussinesq system coupled with a nonlinear heat equation and the $SP_3$ model for radiation. First, we show the existence of unique, regular solutions for this forward system, which is challenging due to the cross-diffusion character of the $SP_3$ model, since it does not allow for the application of standard maximum principles. Second, we show the existence of an optimal boundary control and analyze the linearized state system, which yields the existence of adjoint states and the differentiability of the control-to-state map.

Keywords

natural convection, radiation, SPN-approximation, optimal boundary control, KKT system, adjoints, analysis

2010 Mathematics Subject Classification

35K55, 49K20, 80A20

Full Text (PDF format)