Null-controllability of perturbed porous medium gas flow

Geshkovski, Borjan

doi:10.1051/cocv/2020009

Cited by 8 publications

(7 citation statements)

References 35 publications

(53 reference statements)

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“…x ∈ (0, 1), v(0) = v(1) = 1 (43) As mentioned before, using comparison arguments with sections of traveling waves, we can prove that the solution of the parabolic model converges to 1 for any λ as t → +∞. Proposition 4 (Convergence to 1).…”

Section: Dirichlet Boundary Conditionmentioning

confidence: 81%

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Control of reaction-diffusion models in biology and social sciences

Ruiz-Balet

Zuazua

2022

MCRF

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These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems' dynamics and, in particular, the use of traveling waves and steady-states' paths, can be employed to build controls driving the system to the desired target.We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results.

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Section: Dirichlet Boundary Conditionmentioning

confidence: 81%

“…However, there are several other relevant linear and nonlinear diffusion operators whose analysis is even more challenging. This is the case, for instance, when considering the porous medium equation [119,87,19,43]:…”

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confidence: 99%

Control of reaction-diffusion models in biology and social sciences

Ruiz-Balet

Zuazua

2022

MCRF

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“…The controllability aspects of one-dimensional, parabolic freeboundary problems similar to (1.1) have been addressed in several recent works (see e.g. [13,17,16,19]). In [13,17], Fernández-Cara et al consider the one-phase Stefan problem…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Controllability of One-Dimensional Viscous Free Boundary Flows

Geshkovski¹,

Zuazua²

2021

SIAM J. Control Optim.

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In this work, we address the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. The free boundary manifests itself as one moving end of the interval, and its evolution is given by the value of the fluid velocity at this endpoint. We prove that, by means of a control actuating along the fixed boundary, we may steer the fluid to constant velocity in addition to prescribing the free boundary's position, provided the initial velocities and interface positions are close enough.

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“…Linear convective potentials (as in the first example) may also be added; this allows us to see the framework as linearized Navier-Stokes. Many further examples can be fitted into this framework, including several classes of degenerate linear parabolic equations (Cannarsa, Beauchard and Guglielmi 2013, Gueye 2014, Geshkovski 2020, evolution equations for the fractional Laplacian with Dirichlet boundary conditions (Warma andZamorano 2021, Macià 2021) and so on.…”

Section: Stokes Equationsmentioning

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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Null-controllability of perturbed porous medium gas flow

Cited by 8 publications

References 35 publications

Control of reaction-diffusion models in biology and social sciences

Control of reaction-diffusion models in biology and social sciences

Controllability of One-Dimensional Viscous Free Boundary Flows

Turnpike in optimal control of PDEs, ResNets, and beyond

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