Indirect controllability of locally coupled systems under geometric conditions

Alabau‐Boussouira, Fatiha; Léautaud, Matthieu

doi:10.1016/j.crma.2011.02.004

Cited by 32 publications

(62 citation statements)

References 15 publications

(47 reference statements)

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“…Thus, taking ε ∈ (0, (T − T 0 (p, q))/4), we have the absolute convergence of the series defining v (1) and v (2) in L 2 (0, T ). This ends the proof.…”

Section: Resultsmentioning

confidence: 99%

“…Concerning the null and approximate controllability of systems (1.1) and (1.2) in the case p ≡ 0 and q ≡ 0 in (0, π), a partial answer is given in [1,2,13,23] under the sign condition q 0 or q 0 in (0, π). These results are obtained as a consequence of controllability results of a hyperbolic system using the transmutation method (see [21]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We recall that we look for a control v of the form (3.2) and (3.8) with f (1) and f (2) defined in Lemma 3.5. We will solve the moment problem (3.9) depending on whether k belongs to Λ 1 , Λ 2 or Λ 3 , where…”

Section: 2)mentioning

confidence: 99%

“…Assume that conditions (1.10) and (1.11) hold. Consider the functions f (1) and f (2) defined in Lemma 3.5 and the matrix A 1,k given in (3.10). Let k ∈ N * .…”

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confidence: 99%

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Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Duprez

2017

ESAIM: COCV

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Abstract. This paper is devoted to the controllability of linear systems of two coupled parabolic equations when the coupling involves a space dependent first order term. This system is set on an bounded interval I ⊂⊂ R, and the first equation is controlled by a force supported in a subinterval of I or on the boundary. In the case where the intersection of the coupling and control domains is nonempty, we prove null controllability at any time. Otherwise, we provide a minimal time for null controllability. Finally we give a necessary and sufficient condition for the approximate controllability. The main technical tool for obtaining these results is the moment method.Mathematics Subject Classification. 93B05, 93B07.

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“…Thus, taking ε ∈ (0, (T − T 0 (p, q))/4), we have the absolute convergence of the series defining v (1) and v (2) in L 2 (0, T ). This ends the proof.…”

Section: Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: 2)mentioning

confidence: 99%

“…Assume that conditions (1.10) and (1.11) hold. Consider the functions f (1) and f (2) defined in Lemma 3.5 and the matrix A 1,k given in (3.10). Let k ∈ N * .…”

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confidence: 99%

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Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Duprez

2017

ESAIM: COCV

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“…Thanks to this, we prove observability and controllability results for coupled systems in the case of coercive bounded coupling operators C (case of globally distributed couplings), unbounded control operators (case of boundary control), and in some situations if the diffusion operators are not the same. The results of [1,2] have been recently extended by the author and Léautaud in [3,4] to the case of partially coercive coupling operators (case of localized couplings). The coupling coefficient is assumed to be a sufficiently smooth function.…”

Section: Some Overview On the Literature For Controlled Coupled Systemsmentioning

confidence: 99%

Insensitizing exact controls for the scalar wave equation and exact controllability of $$2$$ -coupled cascade systems of PDE’s by a single control

Alabau‐Boussouira

2013

Math. Control Signals Syst.

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AMS(MOS) subject classifications: 34G10, 35B35, 35B37, 35L90, 93D15, 93D20.Key words and phrases: Boundary observability. Locally distributed observability. Boundary control. Locally distributed control. HUM. Indirect controllability. Hyperbolic systems. Parabolic systems. Schrödinger equations. Cascade systems. Geometric conditions. Abstract linear evolution equations. Insensitizing controls. Optimal conditions. AbstractWe study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE's and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in [2,4] for symmetric coupled systems, to cascade systems which are examples of non symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos Lebeau and Rauch [11]. By duality, this solves the exact controllability, by a single control, of 2-coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger 2-coupled cascade systems under (GCC) and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa [47]. Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by J.-L. Lions [35] and later on by Dáger [19] and Tebou [44].

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Exact controllability of the transmission string–beam equations with a single boundary control

Guo

2023

Math Methods in App Sciences

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We consider the exact controllability of a transmission system which models a flexible beam with one end clamped and the other end linked to a string. The control only acts on the free end of the string. Using the multiplier method, we prove that the full energy of transmission system can be estimated by the boundary data of string. Finally, by using the HUM method, we deduce that the transmission system is exactly controllable at the time of control .

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Indirect controllability of locally coupled systems under geometric conditions

Cited by 32 publications

References 15 publications

Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Insensitizing exact controls for the scalar wave equation and exact controllability of $$2$$ -coupled cascade systems of PDE’s by a single control

Exact controllability of the transmission string–beam equations with a single boundary control

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