partial differential equations; feedback control; reaction-diffusion equations; Pyragas control; time delay; noninvasive control
Noninvasive time-delayed feedback control (''Pyragas control'') has been investigated theoretically, numerically and experimentally during the last twenty years. Its success has been proven or experimentally demonstrated for numerous dynamical systems given by ordinary differential equations.
In this thesis we introduce new noninvasive spatio-temporal control terms for partial differential equations with the purpose to stabilize unstable equilibria and periodic orbits.
We construct these successful control terms by introducing the notion of control triples.
The control triple defines how we transform output signal, space, and time in the control term. This Ansatz, especially well-suited for the control of partial differential equations, does not exist in the literature so far. It incorporates the spatio-temporal patterns of the equilibria and periodic orbits into the control term.
We investigate the new control triple method in the context of scalar reaction-diffusion equations on the circle: For these equations we present two types of control schemes: Control schemes of rotation type combine rotations in space, which we interpret as a spatial delay, with a time delay and a sign change of the output signal, while control schemes of reflection type combine reflections in space, time delay and a sign change of the output signal.
For control schemes of rotation type it turns out that spatial delays of half the spatial period combined with a small time delay and a sign change in the output signal are successful in the stabilization of equilibria and periodic orbits. However, those control terms which use a full spatial period, and consequently no sign change of the output signal, fail their task of stabilization for every time delay. This failure includes the control terms of Pyragas type.
Using control schemes of reflection type, we are able to stabilize orbits with an odd reflection symmetry, but not those with an even symmetry. Here again, the sign change of the output signal decides whether the control is successful or not.
The proof of stabilization uses a modified version of Hill's equation with spatio-temporal delay. We combine Hill's equation with symmetry properties to obtain the results.
Finally, we present a detailed case study for a specific reaction-diffusion equation, namely the Chafee-Infante equation. We discuss possible extensions and limitations of our new control schemes.
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