On a kind of time optimal control problem of the heat equation

In this paper, we deal with time and norm optimal control problems for the heat equation with dynamic boundary conditions. We prove the existence of admissible controls and, combining a suitable Carleman estimate for such an equation with some analyticity results, we show a strong unique continuation property. Then, with the aid of this unique continuation, bang‐bang properties and the connection between time and norm optimal controls can be established.

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“…Proof Using some ideas of Zhang, 30 the proof is carried out in the following three steps: This shows that

\sup_{v \in V_{1}} {true\int}_{0}^{T^{⋆} false(ρ false)} {⟨ e^{(T^{⋆} - s) A} (ρ χ_{ω} v (s), 0), η^{⋆} ⟩}_{𝕃^{2}} d s \leq {true\int}_{0}^{T^{⋆} false(ρ false)} {⟨ e^{(T^{⋆} - s) A} (χ_{ω} v^{⋆} (s), 0), η^{⋆} ⟩}_{𝕃^{2}} d s,

where

V_{1} = : {v : ℝ^{+} \to L^{2} (ω); ‖ v (t) ‖_{L^{2} (ω)} \leq 1, a . e . t \in ℝ^{+}} .

The last inequality is equivalent to

\sup_{v \in V_{1}} {true\int}_{0}^{T^{⋆} false(ρ false)} ⟨ (χ_{ω} ...

…”

Section: Existence and Uniqueness Of Time Optimal Controlmentioning

confidence: 99%

Time and norm optimal controls for the heat equation with dynamical boundary conditions

Boutaayamou

Maniar

Oukdach

2021

Math Methods in App Sciences

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“…Despite 70 years of development, the solution of concrete non-trivial examples of time-optimal control still needs considerable effort [2,4,19]. The problem becomes even more difficult when a control system is described by a partial differential equation [11,24,25,34], particularly, for the heat conductivity equation [12,22,26,29,35,36]. In [1], the correctness of parabolic equations for heat propagation is discussed and for that purpose, a parabolic equation with time delay is considered.…”

Section: Introductionmentioning

confidence: 99%

On the Chernous'ko Time-Optimal Problem for the Equation of Heat Conductivity in a Rod

Азамов

Bakhramov

Akhmedov

2019

UMJ

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The time-optimal problem for the controllable equation of heat conductivity in a rod is considered. By means of the Fourier expansion, the problem reduced to a countable system of one-dimensional control systems with a combined constraint joining control parameters in one relation. In order to improve the time of a suboptimal control constructed by F.L. Chernous'ko, a method of grouping coupled terms of the Fourier expansion of a control function is applied, and a synthesis of the improved suboptimal control is obtained in an explicit form.

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

Maniar

Oukdach

Walid

2023

Differ Equ Dyn Syst

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On a kind of time optimal control problem of the heat equation

Cited by 3 publications

References 33 publications

Time and norm optimal controls for the heat equation with dynamical boundary conditions

Time and norm optimal controls for the heat equation with dynamical boundary conditions

On the Chernous'ko Time-Optimal Problem for the Equation of Heat Conductivity in a Rod

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

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