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Input reconstruction by feedback control for the Schlögl and FitzHugh–Nagumo equations

Maksimov Vyacheslav, Trlötzsch Fredi

International Journal of Applied Mathematics and Computer Science

2020

Vol. 30, no. 1

5--22

Dynamical reconstruction of unknown time-varying controls from inexact measurements of the state function is investigated for a semilinear parabolic equation with memory. This system includes as particular cases the Schlögl model and the FitzHugh–Nagumo equations. A numerical method is suggested that is based on techniques of feedback control. An error analysis is performed. Numerical examples confirm the theoretical predictions.

Optimal sparse boundary control for a semilinear parabolic equation with mixed control-state constraints

Casas Eduardo, Troltzsch Fredi

Control and Cybernetics

2019

Vol. 48, No. 1

89--124

A problem of sparse optimal boundary control for a semilinear parabolic partial diﬀerential equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic objective functional is to be minimized that includes a Tikhonov regularization term and the L1-norm of the control accounting for the sparsity. Applying a recent linearization theorem, we derive ﬁrst-order necessary optimality conditions in terms of a variational inequality under linearized mixed control state constraints. Based on this preliminary result, a Lagrange multiplier rule with bounded and measurable multipliers is derived and sparsity results on the optimal control are demonstrated.

On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints

Neitzel I., Troltzsch F.

Control and Cybernetics

2008

Vol. 37, no 4

1013-1043

Moreau-Yosida and Lavrentiev type regularization methods are considered for nonlinear optimal control problems governed by semilinear parabolic equations with bilateral pointwise control and state constraints. The convergence of optimal controls of the regularized problems is studied for regularization parameters tending to infinity or zero, respectively. In particular, the strong convergence of global and local solutions is addressed. Moreover, strong regularity of the Lavrentiev-regularized optimality system is shown under certain assumptions, which, in particular, allows to show that locally optimal solutions of the Lavrentiev regularized problems are locally unique. This analysis is based on a second-order sufficient optimality condition and a separation assumption on almost active sets.