时滞动力系统中沿周期和拟周期轨道族的Optimization
Optimization along Families of Periodic and Quasiperiodic Orbits in Dynamical Systems with Delay
摘要：
本文推广了一个以前设想的，基于连续优化技术的约束流形上的标量目标函数的情况下，周期和拟周期解的延迟-微分方程。一个拉格朗日形式主义是用来构造伴随条件，是线性和均匀的未知拉格朗日乘子。因此，它示出了如何在约束流形上的临界点，可以通过几个阶段的连续沿一系列的连接一维流形的解决方案，以增加子集的必要最优性条件。由于在原始和伴随微分方程中存在延迟和超前变元，必须注意确定拉格朗日乘子关于时间的光滑度。这样的考虑自然会导致制定多段边值问题（BVP），包括可能性，段的数量可能会改变，或他们的顺序可能置换，在延续。该方法使用的软件包可可的线性和非线性延迟微分方程的周期轨道上说明，请记住，封闭形式的解决方案通常是不可用的，即使在线性的情况下。最后，我们展示了优化的一个家庭的准周期不变环面在一个例子中展开的Hopf分支延迟和参数强迫。准周期的情况下，是一个进一步的原始贡献的文献约束偏微分BVP的优化。
英文摘要：
This paper generalizes a previously-conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package coco on periodic orbits of both linear and nonlinear delay-differential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs.
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