On a Neumann boundary control in a parabolic system

Şener, Sıdıka Şule; Subaşi, Murat

doi:10.1186/s13661-015-0430-5

Cited by 15 publications

(8 citation statements)

References 4 publications

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“…In the paper, we study controllability problems for the heat equation on a half-axis. Note that these problems for the heat equation on domains bounded with respect to spatial variables were investigated rather completely in a number of papers (see, e.g., [3,5,12,16,15,17,18] and references therein). However, controlability problems for the heat equation on domains unbounded with respect to spatial variables have not been fully studied.…”

mentioning

confidence: 99%

Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Fardigola¹,

Khalina²

2021

Mathematical Control &Amp; Related Fields

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In the paper, the problems of controllability and approximate controllability are studied for the control system wt = wxx, wx(0, •) = u, x > 0, t ∈ (0, T ), where u ∈ L ∞ (0, T ) is a control. It is proved that each initial state of the system is approximately controllable to each target state in a given time T . A necessary and sufficient condition for controllability in a given time T is obtained in terms of solvability of a Markov power moment problem. It is also shown that there is no initial state which is null-controllable in a given time T . Orthogonal bases are constructed in H 1 and H 1 . Using these bases, numerical solutions to the approximate controllability problem are obtained. The results are illustrated by examples.

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confidence: 99%

Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Fardigola¹,

Khalina²

2021

Mathematical Control &Amp; Related Fields

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“…Here, we consider the stationary real Neumann boundary optimal control problem (15). If we denote by u ∞bqg to the unique solution of the variational equality (13) for data b, q and g. and if we consider q = λq * 0 for fixed q * 0 ∈ Q (q * 0 = 0) and λ ∈ R, we can see that…”

Section: Real Neumann Boundary Optimal Control Problem In Relation To...mentioning

confidence: 99%

“…Here, since B 2 − 4AC < 0, there exists a unique λ ∞ ∈ R such that satisfies the problem (15), that is…”

Section: Real Neumann Boundary Optimal Control Problem In Relation To...mentioning

confidence: 99%

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Neumann boundary optimal control problems governed by parabolic variational equalities

Bollo¹,

Gariboldi²,

Tarzia³

2021

Preprint

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We consider a heat conduction problem S with mixed boundary conditions in a n-dimensional domain Ω with regular boundary and a family of problems S α with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ 1 . In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ 2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and S α and to mixed elliptic problems P and P α . We find a explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.

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“…In [7] and [8], were studied control problems on the source g and the flux q respectively, for parabolic variational inequalities of second kind. Other papers on the subject are [12,13,14,18,19,20,21,25,26,27]. Our interest is the convergence when α → ∞, which is related to [4,22,23].…”

Section: Introductionmentioning

confidence: 99%

Convergence of simultaneous distributed-boundary parabolic optimal control problems

Tarzia¹,

Bollo²,

Gariboldi³

2020

Evolution Equations &Amp; Control Theory

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We consider a heat conduction problem S with mixed boundary conditions in a n-dimensional domain Ω with regular boundary Γ and a family of problems S α , where the parameter α > 0 is the heat transfer coefficient on the portion of the boundary Γ 1 . In relation to these state systems, we formulate simultaneous distributed-boundary optimal control problems on the internal energy g and the heat flux q on the complementary portion of the boundary Γ 2 . We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient α goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author.

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On a Neumann boundary control in a parabolic system

Cited by 15 publications

References 4 publications

Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Neumann boundary optimal control problems governed by parabolic variational equalities

Convergence of simultaneous distributed-boundary parabolic optimal control problems

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