Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay
Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay
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延迟动力系统中周期和准周期轨道族的优化
DOI：
10.1007/s11071-019-05304-y
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发表时间：
2019
影响因子：
5.6
通讯作者：
Ahsan Z
中科院分区：
文献类型：
--
作者：
Ahsan Z
本文概括了以前设想的，基于连续的优化技术的标量目标函数的约束流形的情况下，延迟微分方程的周期和拟周期解。一个拉格朗日形式主义是用来构建伴随条件，是线性和均匀的未知拉格朗日乘子。因此，它表明如何通过沿着一系列连接的一维流形的解的连续沿着几个阶段找到约束流形上的临界点，以增加必要最优性条件的子集。由于原始和伴随微分方程中存在延迟和提前参数，因此必须小心确定拉格朗日乘数相对于时间的平滑程度。这样的考虑自然会导致制定多段边界值问题（BVP），包括可能性，段的数量可能会改变，或者它们的顺序可能会置换，在延续。该方法使用的软件包ecocoon周期轨道的线性和非线性延迟微分方程，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。
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 packagecocoon 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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影响因子：
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