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Delayed feedback control method for computing the cyclic steady states of evolution problems

机译：演化问题循环稳态的时滞反馈控制方法

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摘要

This work is focused on fast techniques for computing the cyclic steady states of evolution problems in non-linear mechanics with space-time periodicity conditions. In industrial applications, in order to avoid the inversion of very large matrices, such a cyclic solution is usually computed as an asymptotic limit of the associated initial value problem with arbitrary initial data. However, when the relaxation time is high, convergence to the limit cycle can be very slow. In such cases nonetheless, one is not interested in the transient solution, but only in a fast access to the limit cycle. Thus, in this work we modify the problem, introducing the time-delayed feedback control, which is widely used for stabilization of unstable periodic orbits. In our framework it is applied to an initially stable system in order to accelerate its convergence to the limit cycle. Moreover, the control term, based on the space-time periodicity error, includes both shifts in time and in space. Our main result is the optimal form of the control term for a very general class of linear evolution problems, providing the fastest convergence to the cyclic solution, which has been further extended and studied in the non-linear case. Efficiency of the method increases with the problem's relaxation time.
机译：这项工作的重点是快速技术，用于计算具有时空周期性条件的非线性力学中演化问题的循环稳态。在工业应用中，为了避免很大的矩阵求逆，通常将这种循环解计算为具有任意初始数据的关联初始值问题的渐近极限。但是，当弛豫时间较长时，收敛到极限循环可能会非常缓慢。尽管如此，在这种情况下，人们对瞬态解决方案不感兴趣，而只是对快速访问极限循环感兴趣。因此，在这项工作中，我们修改了问题，引入了时滞反馈控制，该控制被广泛用于稳定不稳定的周期性轨道。在我们的框架中，将其应用于最初稳定的系统，以加速其收敛到极限周期。而且，基于时空周期性误差的控制项包括时间和空间上的偏移。我们的主要结果是针对非常普通的一类线性演化问题的控制项的最佳形式，它为循环解提供了最快的收敛性，并且在非线性情况下已经得到了进一步扩展和研究。该方法的效率随着问题的松弛时间而增加。

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