Optimal actuator design via Brunovsky's normal form

Geshkovski, Borjan; Zuazua, Enrique

doi:10.48550/arxiv.2108.05629

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…The problem of characterizing such optimal shapes for linear PDEs has been partially resolved, namely by making T PDE , R N , 253 use of a randomization procedure; see Zuazua (2015, 2016) and the references therein. In the finite-dimensional context, the optimal actuator shape may happen to be time-independent (Geshkovski and Zuazua 2021). But a full picture in the PDE setting is lacking.…”

Section: Turnpike and Optimal Shape Designmentioning

confidence: 99%

Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski

Zuazua²

2022

Acta Numerica

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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 insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the 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: Turnpike and Optimal Shape Designmentioning

confidence: 99%

Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski

Zuazua²

2022

Acta Numerica

Self Cite

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“…The problem of characterizing such optimal shapes for linear PDEs has been partially resolved, namely by making use of a randomization procedure (see [148,149] and the references therein). In the finite dimensional context, the optimal actuator shape may happen to be time-independent ( [69]). But a full picture in the PDE setting is lacking.…”

Section: Part IV Epiloguementioning

confidence: 99%

Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski¹,

Zuazua²

2022

Preprint

Self Cite

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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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“…Taking stock of (7.1), there are a variety of different spectral optimization problems one could then envisage for (1.3), such as characterizing optimal actuator and observer domains ω in the spirit of [24,45,46,47], and in particular, comparing how these designs differ from that of the classical heat equation, or the limit of these designs as σ 0 (should controllability hold for the latter). (3) The obstacle problem.…”

Section: Epiloguementioning

confidence: 99%

Control of the linearized Stefan problem in a periodic box

Geshkovski¹,

Maity²

2022

Preprint

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In this paper we consider the linearized one-phase Stefan problem with surface tension, set in the strip T × (−1, 1), thus with periodic boundary conditions respect to the horizontal direction x1 ∈ T. When the support of the control is not localized in x1, namely, is of the form ω = T × (c, d), we prove that the system is null-controllable in any positive time. We rely on a Fourier decomposition with respect to x1, and controllability results which are uniform with respect to the Fourier frequency parameter for the resulting family of one-dimensional systems. The latter results are also novel, as we compute the full spectrum of the underlying operator for the non-zero Fourier modes. The zeroth mode system, on the other hand, is seen as a controllability problem for the linear heat equation with a finite-dimensional constraint. We extend the controllability result to the setting of controls with a support localized in a box: ω = (a, b) × (c, d), through an argument inspired by the method of Lebeau and Robbiano, under the assumption that the initial data are of zero mean. Numerical experiments motivate several challenging open problems, foraying even beyond the specific setting we deal with herein.

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Optimal actuator design via Brunovsky's normal form

Cited by 3 publications

References 13 publications

Turnpike in optimal control of PDEs, ResNets, and beyond

Turnpike in optimal control of PDEs, ResNets, and beyond

Turnpike in optimal control of PDEs, ResNets, and beyond

Control of the linearized Stefan problem in a periodic box

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