Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs

Hernández-Santamaría, Víctor; Teresa, Luz de

doi:10.1007/978-3-319-97613-6_7

Cited by 12 publications

(8 citation statements)

References 20 publications

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“…To prove the needed observability inequality, we are going to improve the Carleman inequality (28). To this end, following [5], we modify the weight functions ξ and α.…”

Section: Carleman Estimatesmentioning

confidence: 99%

“…Proof. The proof relies on the Carleman estimate (28) and some energy estimates. Following the strategy in [5], let us introduce a function η ∈ C 1 ([0, T ]) such that…”

Section: Carleman Estimatesmentioning

confidence: 99%

“…Assume that (10) holds. Then there exist λ3 ≥ 1, s3 ≥ 1 and C > 0 such that every solution (Z, Ψ 1 , Ψ 2 ) to (26) satisfies I(s, λ; Z) + I(s, λ; H) ≤ Cs 7 λ 8 ω×(0,T ) e −2sα ξ 7 |z| 2 dx dt(28) …”

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confidence: 99%

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Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

Boutaayamou,

Maniar,

Oukdach

2021

Preprint

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This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.

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“…To prove the needed observability inequality, we are going to improve the Carleman inequality (28). To this end, following [5], we modify the weight functions ξ and α.…”

Section: Carleman Estimatesmentioning

confidence: 99%

“…Proof. The proof relies on the Carleman estimate (28) and some energy estimates. Following the strategy in [5], let us introduce a function η ∈ C 1 ([0, T ]) such that…”

Section: Carleman Estimatesmentioning

confidence: 99%

See 1 more Smart Citation

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

Boutaayamou,

Maniar,

Oukdach

2021

Preprint

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show abstract

“…More recently, in [8], the authors dealt with the hierarchical exact controllability of parabolic equations with distributed and boundary controls. The same method was also applied in the context of parabolic coupled systems in [29] and [27] and mixed with robust control in [28] and [39]. The previous works have considered Dirichlet and Neumann boundary conditions.…”

mentioning

confidence: 99%

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

Boutaayamou¹,

Maniar²,

Oukdach³

2022

EECT

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<p style='text-indent:20px;'>This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.</p>

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“…(5.8) This means that we have to control one component of a system of four coupled equations. As pointed out in [25], the hierarchic controllability of coupled systems as (5.8) with one control force is not yet fully understood and further investigation is desirable.…”

mentioning

confidence: 99%

Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary

Hernández-Santamaría¹,

Peralta²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we present some controllability results for the heat equation in the framework of hierarchic control. We present a Stackelberg strategy combining the concept of controllability with robustness: the main control (the leader) is in charge of a null-controllability objective while a secondary control (the follower) solves a robust control problem, this is, we look for an optimal control in the presence of the worst disturbance. We improve previous results by considering that either the leader or follower control acts on a small part of the boundary. We also present a discussion about the possibility and limitations of placing all the involved controls on the boundary.

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Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs

Cited by 12 publications

References 20 publications

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary

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