Abstract
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.
Highlights
The optimization of time-delay systems has been the subject of intensive research for many years
Λ f (τ ) and λbc are the Lagrange multipliers associated with the imposition of the differential equations and boundary conditions, respectively, and each integrand is assumed to be continuously differentiable on the corresponding interval
The vector-valued functions λ f (φ, τ ) and λrot(φ), and the scalar λph are the Lagrange multipliers associated with the imposition of the differential equations (55) and
Summary
The optimization of time-delay systems has been the subject of intensive research for many years. Liao et al [13] developed an optimization technique for periodic solutions of delay-differential equations using the harmonic balance method and continuation techniques They posed an amplitude optimization problem subject to the algebraic constraints obtained by substitution of a truncated Fourier representation in the governing equation along with the stability conditions. There, a sequence of properly initialized stages of continuation along onedimensional manifolds of solutions to a subset of the necessary optimality conditions was used to connect the local extremum to an initial solution guess with vanishing Lagrange multipliers This methodology was recently revisited by Li and Dankowicz [12] and there cast in terms of partial Lagrangians relevant to the general context of constrained optimization of integrodifferential boundary-value problems without delay. A number of additional considerations and opportunities for future work are considered in the concluding section
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