On the optimal control of the Schlögl-model

• Published in: Computational optimization and applications Vol. 56; no. 1; pp. 153 - 185
• Main Authors: Buchholz, Rico, Engel, Harald, Kammann, Eileen, Tröltzsch, Fredi
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-09-2013; Springer Nature B.V
• Subjects: Analysis; Calculus; Conjugate gradient method; Convex and Discrete Geometry; Management Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinearity; Numbers; Numerical analysis; Operations Research; Operations Research/Decision Theory; Optimal control; Optimization; Partial differential equations; Physics; Statistics; Studies; Travelling wave front; Model reduction; Semilinear parabolic equation; Optimal control; Schlögl model
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-013-9550-y

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation.
The well-posedness of the optimal control problem and the regularity of its solution are proved. First-order necessary optimality conditions are established by standard adjoint calculus. The state equation is solved by the implicit Euler method in time and a finite element technique with respect to the spatial variable. Moreover, model reduction by Proper Orthogonal Decomposition is applied and compared with the numerical solution of the full problem. To solve the optimal control problems numerically, the performance of different versions of the nonlinear conjugate gradient method is studied. Various numerical examples demonstrate the capacities and limits of optimal control methods.