The Inverse Problem for Hamilton--Jacobi Equations and Semiconcave Envelopes

Esteve, Carlos; Zuazua, Enrique

doi:10.1137/20m1330130

Cited by 14 publications

(11 citation statements)

References 17 publications

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“…In [19], the authors prove that the set of initial data evolving to an attainable target u T is a convex set. Later on, the aforementioned set was fully characterized in [12], [16] using the classical Lax-Hopf formula [21,Theorem 2.1].…”

Section: B Presentation Of the Problemmentioning

confidence: 99%

“…In our paper, an alternative proof of the characterization of the set of initial data in [12], [16] is given using only backward generalized characteristics. This leads to the hope of investigate systems of conservation laws in one dimension.…”

Section: B Presentation Of the Problemmentioning

confidence: 99%

“…Due to the property of non-backward uniqueness of Burgers equation, there may exist multiple initial data leading to the same given target. In [12], [16], the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given.…”

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confidence: 99%

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Initial Data Identification for the One-Dimensional Burgers Equation

Liard

Zuazua²

2022

IEEE Trans. Automat. Contr.

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In this paper, we study the problem of identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of non-backward uniqueness of Burgers equation, there may exist multiple initial data leading to the same given target. In [12], [16], the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigate systems of conservation laws in one dimension where the classical Lax-Hopf formula doesn't hold anymore. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wave-front tracking algorithm.

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Section: B Presentation Of the Problemmentioning

confidence: 99%

Section: B Presentation Of the Problemmentioning

confidence: 99%

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confidence: 99%

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Initial Data Identification for the One-Dimensional Burgers Equation

Liard

Zuazua²

2022

IEEE Trans. Automat. Contr.

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“…In Theorem 1.5, we perform a sort of inverse analysis for the Hamilton-Jacobi flow, in the sense that we determine some features of initial data starting from properties of the corresponding flows. As for another type of inverse problems for Hamilton-Jacobi equations, see [4,13] etc.…”

Section: Introductionmentioning

confidence: 99%

Hamilton–Jacobi flows with nowhere differentiable initial data

2022

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We deal with a sort of inverse problem. Namely, we aim at deriving features of initial data of an Hamilton-Jacobi flow starting from specific assumptions on the flow. To this scope, we introduce a class of nowhere differentiable functions, and characterize a function in this class through a property of the Hamilton–Jacobi flow issued from this function. This analysis shows that, despite the regularizing effect of the flow, nowhere differentiability of initial data can be nevertheless detected looking at the flow for positive times.

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“…In some settings, the mathematical formulation of the sonic-boom minimization problem can be seen as an inverse design or optimal control problem for the inviscid Burgers equation over long time horizons ( [7,5]). The stability and inversion properties of such problems are sensitive to the time horizon and analytical guarantees are thus required ( [121,52]). Similar considerations transfer to optimal control problems in climate science ( [136,106]), or data assimilation problems in meteorology and oceanography ( [70]), in addition to problems transversing fields such as the computation of sensitivities, all of which fit within the framework discussed above.…”

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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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The Inverse Problem for Hamilton--Jacobi Equations and Semiconcave Envelopes

Cited by 14 publications

References 17 publications

Initial Data Identification for the One-Dimensional Burgers Equation

Initial Data Identification for the One-Dimensional Burgers Equation

Hamilton–Jacobi flows with nowhere differentiable initial data

Turnpike in optimal control of PDEs, ResNets, and beyond

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