Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

Heiland, Jan; Zuazua, Enrique

doi:10.48550/arxiv.2007.13621

Cited by 2 publications

(2 citation statements)

References 25 publications

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“…In the finite dimensional case, dynamical systems techniques based on stable manifold theory have also been used and developed for proving the exponential turnpike property ( [159]). Further links with systems theory are established in [95], and additional direct strategies for proving exponential turnpike properties for finite dimensional systems may be found in [128].…”

Section: 4mentioning

confidence: 99%

Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski¹,

Zuazua²

2022

Preprint

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The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade -the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal-, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.

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Section: 4mentioning

confidence: 99%

Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski¹,

Zuazua²

2022

Preprint

Self Cite

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“…The turnpike phenomenon is usually analyzed in two different situations: (a) supposing that the OCP is regular allows transcribing the first-order optimality conditions as a system ODEs, cf. [18,19,29,34]; (b) supposing the underlying system, respectively, the OCP as such is strictly dissipative with respect to a specific steady state, cf. [8,14,17,33].…”

Section: Introductionmentioning

confidence: 99%

Optimal control of port-Hamiltonian descriptor systems with minimal energy supply

Faulwasser¹,

Maschke²,

Philipp³

et al. 2021

Preprint

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We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. Further, we derive a input-state turnpike towards a subspace for optimal control of port-Hamiltonian ordinary differential equations with a feed-through term and a turnpike property for the corresponding adjoint states towards zero. In an appendix we characterize the class of dissipative Hamiltonian matrices and pencils.

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Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

Cited by 2 publications

References 25 publications

Turnpike in optimal control of PDEs, ResNets, and beyond

Turnpike in optimal control of PDEs, ResNets, and beyond

Optimal control of port-Hamiltonian descriptor systems with minimal energy supply

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