4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

Chekroun, Mickaël D.; Kröner, Axel; Liu, Honghu

doi:10.1515/9783110543599-004

Cited by 3 publications

(2 citation statements)

References 54 publications

(107 reference statements)

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“…The interested reader is referred to [CKL17] for more general cases, and also for convergence results concerning the value functions. Applications to control in feedback form can be found in [CKL18].…”

Section: Control From Effective Reduced Galerkin Systemsmentioning

confidence: 99%

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Optimal management of harvested population at the edge of extinction

Chekroun,

Liu

2020

Preprint

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A. Optimal control of harvested population at the edge of extinction in an unprotected area, is considered. The underlying population dynamics is governed by a Kolmogorov-Petrovsky-Piskunov equation with a harvesting term and space-dependent coefficients while the control consists of transporting individuals from a natural reserve. The nonlinear optimal control problem is approximated by means of a Galerkin scheme. Convergence result about the optimal controlled solutions and error estimates between the corresponding optimal controls, are derived. For certain parameter regimes, nearly optimal solutions are calculated from a simple logistic ordinary differential equation (ODE) with a harvesting term, obtained as a Galerkin approximation of the original partial differential equation (PDE) model. A critical allowable fraction α of the reserve's population is inferred from the reduced logistic ODE with a harvesting term. This estimate obtained from the reduced model allows us to distinguish sharply between survival and extinction for the full PDE itself, and thus to declare whether a control strategy leads to success or failure for the corresponding rescue operation while ensuring survival in the reserve's population. In dynamical terms, this result illustrates that although continuous dependence on the forcing may hold on finite-time intervals, a high sensitivity in the system's response may occur in the asymptotic time. We believe that this work, by its generality, establishes bridges interesting to explore between optimal control problems of ODEs with a harvesting term and their PDE counterpart.

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Section: Control From Effective Reduced Galerkin Systemsmentioning

confidence: 99%

“…[CH98,Tem97] for such a priori bounds for nonlinear partial differential equations. Such bounds can also be derived for nonlinear systems of delay differential equations (DDEs); see [CGLW16,CKL18]…”

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Optimal management of harvested population at the edge of extinction

Chekroun,

Liu

2020

Preprint

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Control of Multi-Agent Systems with Distributed Delay

Márton

2023

IFAC-PapersOnLine

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Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models

Chekroun

Koren

Liu

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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By means of Galerkin–Koornwinder (GK) approximations, an efficient reduction approach to the Stuart–Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of delay differential equations that covers the cases of discrete as well as distributed delays. The focus is on the Hopf bifurcation as a consequence of the critical equilibrium’s destabilization resulting from an eigenpair crossing the imaginary axis. The nature of the resulting Hopf bifurcation (super- or subcritical) is then characterized by the inspection of a Lyapunov coefficient easy to determine based on the model’s coefficients and delay parameters. We believe that our approach, which does not rely too much on functional analysis considerations but more on analytic calculations, is suitable to concrete situations arising in physics applications. Thus, using this GK approach to the Lyapunov coefficient and the SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand and Koren, Tziperman, and Feingold on the other are analyzed. Noteworthy is the existence of the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined, in particular, by the intensity of the KF model’s nonlinear effects. “Islands” of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation “sea”; these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues.

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4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

Cited by 3 publications

References 54 publications

Optimal management of harvested population at the edge of extinction

Optimal management of harvested population at the edge of extinction

Control of Multi-Agent Systems with Distributed Delay

Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models

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