Constrained control of gene-flow models

Mazari, Idriss; Ruiz-Balet, Domènec; Zuazua, Enrique

doi:10.48550/arxiv.2005.09236

Cited by 4 publications

(7 citation statements)

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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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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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“…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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“…Evidently, controlling the final configuration of a system is a fundamental feature among several aspects characterizing the system, for instance, to avoid the extinction of a population or to cease the propagation of an infectious disease. A complete description of this problem with several applications in different areas can be seen in the following works [11,12,13,15].…”

Section: Introductionmentioning

confidence: 99%

“…The above definition is slightly different from the one commonly used ( [11,12,13]) since we are only interested in static controls, for this reason we add the term "static" in the definition of controllability.…”

Section: Introductionmentioning

confidence: 99%

“…Dirichlet static controls have been used successfully on extinction results (when the target is the solution y ≡ 0), invasion (y ≡ 1) or a state of coexistence (y ≡ θ for the bistable case). It is important to note the findings of [13], where an identical idea of ensuring the uniqueness of the desired target was successfully implemented in multidimensional models, and in [11] for problems dealing with drift term (related to the gene-flow phenomenon). The main objective in these works is to control the problem towards homogeneous steady-states y ≡ 0, 1 or θ (for the bistable case).…”

Section: Introductionmentioning

confidence: 99%

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A note on control of one-dimensional heterogeneous reaction-diffusion equations

Sônego

Roychowdhury

2023

EECT

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<p style='text-indent:20px;'>We are concerned with the controllability of a general heterogeneous semilinear parabolic equation of the form <inline-formula><tex-math id="M1">\begin{document}$ y_t(x,t)=y_{xx}(x,t)+f(x,y(x,t)) $\end{document}</tex-math></inline-formula> in one space dimension using boundary static controls. More precisely, we provide conditions so that, using only static controls at the boundary, a certain target (a steady-state of the equation) is reached in infinite time. Applications of the main result are also presented, in particular, we provide several controllability results to heterogeneous equations of monostable and bistable types.</p>

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Constrained control of gene-flow models

Cited by 4 publications

References 0 publications

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

A note on control of one-dimensional heterogeneous reaction-diffusion equations

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