Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

Ruiz-Balet, Domènec; Zuazua, Enrique

doi:10.1016/j.matpur.2020.08.006

Cited by 16 publications

(29 citation statements)

References 23 publications

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“…The proofs of Theorems 2.6 and 2.8, presented in Section 3, are based on a "staircase" strategy, that has proven its efficiency for the study of state-constrained or control-constrained controllability [3,5,8,11]. The idea is to make small steps towards the target, following a path of trajectories such that the controlled trajectory stays always close to a nonnegative free trajectory, and therefore almost nonnegative (see Figures 1 and 2).…”

Section: Results On State-constrained Controllabilitymentioning

confidence: 99%

“…State-constrained controllability is a challenging subject that has gained popularity in the last few years, notably at the instigation of Jérome Lohéac, Emmanuel Trélat and Enrique Zuazua in the seminal paper [2], in which some controllability results with positivity constraints on the state or the control for the linear heat equation are proved, under a minimal time condition which turns out to be necessary. This question yielded to an increasing number of articles in different frameworks, many of them being coauthored by Enrique Zuazua: for ODE systems [3,4], semilinear and quasilinear heat equations [5,6], monostable and bistable reaction-diffusion equations [7,8], the fractional one-dimensional heat equation [9,10], wave equations [11], and age-structured systems [12]. The spirit of most of these results can be summarized this way: when the considered system is controllable in the classical sense, and sometimes under assumptions on the initial and target states or on the system properties, controllability with a constraint on the state is possible with a positive minimal time.…”

Section: Introductionmentioning

confidence: 99%

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State-constrained controllability of linear reaction-diffusion systems

Lissy

Moreau

2021

ESAIM: COCV

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We study the controllability of a coupled system of linear parabolic equations, with nonnegativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an “approximate” nonnegativity constraint, and a another stronger one, with “exact” nonnegativity constraint, when all the diffusion coefficients are equal. The proofs are based on a “staircase” method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state.

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Section: Results On State-constrained Controllabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

State-constrained controllability of linear reaction-diffusion systems

Lissy

Moreau

2021

ESAIM: COCV

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“…Although the controllability theory under constraints on the state and/or control for linear and semilinear PDEs is now rather well established ( [126,141,94,142,146,130,113,157,125,132]), the validity and proof of the exponential turnpike property in such cases is a challenging open problem. 15.6.…”

Section: Part IV Epiloguementioning

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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“…For control purposes, one could envisage transferring results from the Stefan problem to the parabolic obstacle problem (which is actually a problem to be studied in its own right, [20,51]). But this is highly nontrivial due to the fact that the Duvaut transform applies to non-negative solutions of the Stefan problem, and it is not clear if existing techniques on controllability under positivity constraints ( [39], or the so-called staircase method [41,44,52]) would be applicable here…”

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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Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

Cited by 16 publications

References 23 publications

State-constrained controllability of linear reaction-diffusion systems

State-constrained controllability of linear reaction-diffusion systems

Turnpike in optimal control of PDEs, ResNets, and beyond

Control of the linearized Stefan problem in a periodic box

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