SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Trust-region proper orthogonal decomposition for flow control
Trust-region proper orthogonal decomposition for flow control
Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent
ACM Transactions on Mathematical Software (TOMS)
Handbook of Chaos Control
SIAM Journal on Optimization
Nonlinear Model Reduction via Discrete Empirical Interpolation
SIAM Journal on Scientific Computing
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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.