Global boundary null-controllability of one-dimensional semilinear heat equations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00586683" target="_blank" >RIV/67985840:_____/24:00586683 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.3934/dcdss.2024003" target="_blank" >https://doi.org/10.3934/dcdss.2024003</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.3934/dcdss.2024003" target="_blank" >10.3934/dcdss.2024003</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Global boundary null-controllability of one-dimensional semilinear heat equations

Popis výsledku v původním jazyce

This paper addresses the boundary null-controllability of the semilinear heat equation ∂ty - ∂xxy + f(y) = 0, (x, t) ∈ (0, 1) × (0, T). Assuming that the function f ∈ C1(R) satisfies lim sup|r|→+∞ |f(r)|/(|r| ln3/2 |r|) ≤ β for some β > 0 small enough and that the initial datum belongs to L∞(0, 1), we prove the global null-controllability using the Schauder fixed point theorem and a linearization for which the term f(y) is seen as a right side of the equation. Then, assuming that f satisfies lim sup|r|→∞ |f'(r)|/ ln3/2 |r| ≤ β for some β small enough, we show that the fixed point application is contracting yielding a constructive method to approximate boundary controls for the semilinear equation. The crucial technical point is a regularity property of a state-control pair for a linear heat equation with L2 right hand side obtained by using a global Carleman estimate with boundary observation. Numerical experiments illustrate the results. The arguments developed can notably be extended to the multi-dimensional case.

Název v anglickém jazyce

Global boundary null-controllability of one-dimensional semilinear heat equations

Popis výsledku anglicky

This paper addresses the boundary null-controllability of the semilinear heat equation ∂ty - ∂xxy + f(y) = 0, (x, t) ∈ (0, 1) × (0, T). Assuming that the function f ∈ C1(R) satisfies lim sup|r|→+∞ |f(r)|/(|r| ln3/2 |r|) ≤ β for some β > 0 small enough and that the initial datum belongs to L∞(0, 1), we prove the global null-controllability using the Schauder fixed point theorem and a linearization for which the term f(y) is seen as a right side of the equation. Then, assuming that f satisfies lim sup|r|→∞ |f'(r)|/ ln3/2 |r| ≤ β for some β small enough, we show that the fixed point application is contracting yielding a constructive method to approximate boundary controls for the semilinear equation. The crucial technical point is a regularity property of a state-control pair for a linear heat equation with L2 right hand side obtained by using a global Carleman estimate with boundary observation. Numerical experiments illustrate the results. The arguments developed can notably be extended to the multi-dimensional case.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Discrete and Continuous Dynamical systems - Series S

ISSN

1937-1632

e-ISSN

1937-1179

Svazek periodika

17

Číslo periodika v rámci svazku

7

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

47

Strana od-do

2251-2297

Kód UT WoS článku

001150177100001

EID výsledku v databázi Scopus

2-s2.0-85195043592