Lebeau–Robbiano Inequality for Heat Equation with Dynamic Boundary Conditions and Optimal Null Controllability

The paper deals with time and norm optimal control problems for the heat equation with dynamic boundary conditions in the context of the null controllability. More precisely, we answer an open question left in our previous paper (Boutaayamou et al., in Math Methods Appl Sci 45:1359–1376, 2021). To d...

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Bibliographic Details
Published inDifferential equations and dynamical systems Vol. 33; no. 2; pp. 401 - 417
Main Authors Maniar, Lahcen, Oukdach, Omar, Zouhair, Walid
Format Journal Article
LanguageEnglish
Published New Delhi Springer India 01.04.2025
Springer Nature B.V
Subjects
Boundary conditions
Computer Science
Controllability
Convexity
Differential equations
Engineering
Mathematics
Mathematics and Statistics
Observability (systems)
Optimal control
Original Research
Thermodynamics
Logarithmic convexity
Bang–Bang properties
Secondary 93C27
Time and norm optimal controls
Primary 35R12
Dynamic boundary conditions
49N25
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Summary:The paper deals with time and norm optimal control problems for the heat equation with dynamic boundary conditions in the context of the null controllability. More precisely, we answer an open question left in our previous paper (Boutaayamou et al., in Math Methods Appl Sci 45:1359–1376, 2021). To do so, we first prove a new Lebeau–Robbiano spectral inequality using a logarithmic convexity inequality, then an observability inequality on any set of positive measure. This plays a relevant role in proving the existence and uniqueness of optimal null controls. Finally, the connection between time and norm optimal null controls is presented.
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ISSN:0971-3514
0974-6870
DOI:10.1007/s12591-023-00633-2