Maximal Controllability for Boundary Control Problems

Engel, Klaus‐Jochen; Fijavž, Marjeta Kramar; Kloss, B. M.; Nagel, Rainer; Sikolya, Eszter

doi:10.1007/s00245-010-9101-1

Cited by 37 publications

(45 citation statements)

References 32 publications

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“…(b)⇒(a) DefineB t 0 ∈ L(L p ([0, t 0 ], ∂X), X −1 ) by the right-hand-side of (A.4) for t = t 0 . Then by[16, Prop. 2.7] we haveBt 0 W 2,p 0 ([0,t 0 ],∂X) = B t 0 W 2,p 0 ([0,t 0 ],∂X) .…”

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confidence: 96%

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Waves and diffusion on metric graphs with general vertex conditions

Engel¹,

Fijavž²

2019

Evolution Equations &Amp; Control Theory

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We prove well-posedness for general linear wave-and diffusion equations on compact or non-compact metric graphs allowing various conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on2010 Mathematics Subject Classification. 47D06, 35R02, 35G46, 35K05, 35L05. 1 xn ⊺ .Using this notation we define the transformationsJ ϕ e f e ∶= f e ○ ϕ e for f e ∈ X e , Jφi ∈ L(X i ),Then J ϕ e and Jφi are invertible with bounded inverses J −1 ϕ e = J (ϕ e ) −1 ∈ L(X e ) and J −1 ϕ i = J (φ i ) −1 ∈ L(X i ). These maps will be used in Lemma 2.4 to transform space dependent diffusion coefficients into constant ones.

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“…3.16] this implies that(B t ) t∈[0,t 0 ] ⊂ L(L p ([0, t 0 ], ∂X), X) is strongly continuous. Finally, by[16, Prop. 2.8] for u ∈ W 2,p 0 ([0, t 0 ], ∂X) the function x(t) ∶= B t u gives the unique classical solution of (A.3).…”

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confidence: 98%

Waves and diffusion on metric graphs with general vertex conditions

Engel¹,

Fijavž²

2019

Evolution Equations &Amp; Control Theory

Self Cite

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“…The reason for such an active interest in this area is that the range of application is vast. We mention here only a selection from the most recent ones: boundary perturbations by Nickel [29], boundary feedback by Casarino, Engel, Nagel and Nickel [6], boundary control by Engel, Kramar Fijavž, Klöss, Nagel and Sikolya [11] and Engel and Kramar Fijavž [10], port‐Hamiltonian systems by Baroun and Jacob [3], control theory by Jacob, Nabiullin, Partington and Schwenninger [19, 20] and Jacob, Schwenninger and Zwart [21] and vertex control in networks by Engel and Kramar Fijavž [9, 12].…”

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confidence: 99%

A Desch–Schappacher perturbation theorem for bi‐continuous semigroups

Budde

Farkas

2020

Mathematische Nachrichten

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We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra ℒ( ) of bounded, linear operators on a Banach space , and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous semiflows. For both of these classes we present an application of our abstract perturbation theorem. K E Y W O R D Sbi-continuous semigroups, Desch-Schappacher type perturbation, extrapolation spaces, implemented semigroups M S C ( 2 0 1 0 ) 34G10, 46A70, 47A55, 47D03 INTRODUCTIONAs suggested by Greiner in [17] abstract perturbation theory of one-parameter semigroups provides good means to change the domain of a semigroup generator. For this an enlargement of the underlying Banach space may be necessary and extrapolation spaces become important. One of the well-known results in this direction goes back to the papers of Desch and Schappacher, see [7] and [8]. Another prominent example of such general perturbation techniques is due to Staffans and Weiss, [30,31], and an elegant abstract operator theoretic/algebraic approach has been developed by Adler, Bombieri and Engel in [1]. A general theory of unbounded domain perturbations is given by Hadd, Manzo and Rhandi [18]. A more recent paper by Bátkai, Jacob, Voigt and Wintermayr [4] extends the notion of positivity to extrapolation spaces, and studies positive perturbations for positive semigroups on AM-spaces. Hence, the study of abstract Desch-Schappacher type perturbations is a lively research field, to which we contribute with the present article. The reason for such an active interest in this area is that the range of application is vast. We mention here only a selection from the most recent ones: boundary perturbations by Nickel [29], boundary feedback by Casarino, Engel, Nagel and Nickel [6], boundary control by Engel, Kramar Fijavž, Klöss, Nagel and Sikolya [11] and Engel and Kramar Fijavž [10], port-Hamiltonian systems by Baroun and Jacob [3], control theory by Jacob, Nabiullin, Partington and Schwenninger [19,20] and Jacob, Schwenninger and Zwart [21] and vertex control in networks by Engel and Kramar Fijavž [9,12].All the previously mentioned abstract perturbation results were developed for strongly continuous semigroups of linear operators on Banach spaces, 0 -semigroups for short. This is, for certain applications, e.g., for the theory of Markov transition semigroups (see [27], for a more general theory we refer to [26]), far too restrictive. For this situation the Banach space of

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“…Introduction. This paper is a continuation of [16,17] where we introduced a semigroup approach to boundary control problems and applied it to the control of flows in networks. While in these previous works we concentrated on maximal approximate controllability, we now focus on the exact-and positive controllability spaces.…”

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confidence: 99%