Optimization of a model Fokker-Planck equation. (English) Zbl 1255.85018

Herty, Michael; Jörres, Christian; Sandjo, Albert N.

Kinet. Relat. Models 5, No. 3, 485-503 (2012).

Summary: We discuss optimal control problems for the Fokker–Planck equation arising in radiotherapy treatment planning. We prove existence and uniqueness of an optimal boundary control for a general tracking–type cost functional in three spatial dimensions. Under additional regularity assumptions we prove existence of a continuous necessary first–order optimality system. In the one–dimensional case we analyse a numerical discretization of the Fokker–Planck equation. We prove that the resulting discrete optimality system is a suitable discretization of the continuous first–order system.

MSC:

85A25	Radiative transfer in astronomy and astrophysics
76P05	Rarefied gas flows, Boltzmann equation in fluid mechanics
49J20	Existence theories for optimal control problems involving partial differential equations
92C50	Medical applications (general)

Keywords:

Fokker-Planck equation; optimal control; \(P_N\)-approximation