On the optimal control of the Schlögl-model
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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.
KeywordsSchlögl model Semilinear parabolic equation Travelling wave front Optimal control Model reduction
The authors are very grateful to Christopher Ryll (TU Berlin, Institute of Mathematics), who updated and partially improved the numerical results of the third author after she finished her university career.
- 2.Arian, E., Fahl, M., Sachs, E.W.: Trust-region proper orthogonal decomposition for flow control. Tech. rep., ICASE (2000). Technical Report 2000-25 Google Scholar
- 9.Fahl, M., Sachs, E.W.: Reduced order modelling approaches to PDE-constrained optimization based on proper orthogonal decomposition. In: Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B. (eds.) Large-Scale PDE-Constrained Optimization, vol. 30, pp. 268–280. Springer, Berlin (2003) CrossRefGoogle Scholar
- 10.Hager, W.W., Zhang, H.: CG DESCENT, a conjugate gradient method with guaranteed descent. Tech. rep., ACM TOMS (2006) Google Scholar
- 18.Kunisch, K., Wagner, M.: Optimal control of the bidomain system (II): Uniqueness and regularity theorems for weak solutions. Ann. Mat. Pura Appl. doi: 10.1007/s10231-012-0254-1
- 19.Kunisch, K., Wagner, M.: Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions. ESAIM: Math. Model. Numer. Anal. doi: 10.1051/m2an/2012058
- 23.Löber, J.: Nonlinear excitation waves in spatially heterogenous reaction-diffusion systems. Tech. rep, TU Berlin, Institute of Theoretical Physics (2009) Google Scholar
- 25.Müller, M.: Uniform convergence of the POD method and applications to optimal control. PhD thesis, Johannes Kepler University, Graz (2011) Google Scholar
- 32.Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture notes, Institute of Mathematics and Scientific Computing, University of Graz (2007) Google Scholar