Applied Functional Analysis and Partial Differential Equations

Miklavčič, Milan

doi:10.1142/9789812796233

Cited by 36 publications

(57 citation statements)

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“…The operator A is closed, and thus the above properties imply (see, e.g., [30, Theorem 3.7.12], [19,Theorem 4…”

Section: Proof Of the Balanced Realization In Theoremmentioning

confidence: 99%

“…Then the operator A generates an analytic C 0 -semigroup (see, e.g., [19,Section 4.5]), which we assume is exponentially stable. The adjoint operator A * : D(A * ) ⊂ X → X can also be derived from the sesquilinear form:…”

Section: Parabolic Systemsmentioning

confidence: 99%

“…Let be any element of X * . The Bochner integral satisfies (see, e.g., [19,Theorem 4 We now use scalar-valued integration by parts:…”

Section: Proof Of the Balanced Realization In Theoremmentioning

confidence: 99%

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Balanced POD for model reduction of linear PDE systems: convergence theory

Singler

2011

Numer. Math.

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We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15 (3): [997][998][999][1000][1001][1002][1003][1004][1005][1006][1007][1008][1009][1010][1011][1012][1013] 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For exponentially stable systems with bounded finite rank input and output operators, we prove that the balanced realization can be expressed in terms of balancing modes, which are related to the Hankel operator. The balancing modes are required to be smooth, and this can cause computational difficulties for PDE systems. We show how this smoothness requirement can be lessened for parabolic systems, and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems. We prove entrywise convergence of the matrices in the approximate reduced order models in both cases, and present numerical results for two example PDE systems. Mathematics Subject Classification (2000)MSC 65 · MSC 93 · MSC 35

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“…The operator A is closed, and thus the above properties imply (see, e.g., [30, Theorem 3.7.12], [19,Theorem 4…”

Section: Proof Of the Balanced Realization In Theoremmentioning

confidence: 99%

Section: Parabolic Systemsmentioning

confidence: 99%

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Balanced POD for model reduction of linear PDE systems: convergence theory

Singler

2011

Numer. Math.

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“…From the theory of analytic semigroups it is well-known that for sufficiently smooth right-hand side f (t) the solution u(t) also enjoys a certain degree of smoothness for t > 0; cf., e.g. [12]. More specifically, we assume f ∈ C 5 ([0, T ], H) and that f is consistent with the initial data u 0 in such a way that u ∈ C 6 ([0, T ], H).…”

Section: Asymptotic Error Expansion For the Bdf 2 Schemementioning

confidence: 99%

On the stability and error structure of BDF schemes applied to linear parabolic evolution equations

Auzinger

Kramer

2010

Bit Numer Math

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We continue the work of various authors on the stability and error structure of multistep schemes applied to linear evolution equations. BDF schemes are considered, and, as far as reasonable, explicit expressions for all occurring bounds are specified, exploiting prior work on the location of characteristic roots. The 2-step BDF scheme is considered in particular detail, and for problems of sectorial type, an asymptotic error expansion is derived based on damping properties of the scheme.

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“…where A is elliptic second-order linear differential operator with smooth coefficients that do not depend on t and generates a strongly continuous semi-group 0 ( ( )) t S t ≥ on the Hilbert state space 2 (Ω), L the reader can be referred to ( [5,8] We recall some definitions related to regional boundary controllability [9].…”

Section: Problem Statementmentioning

confidence: 99%

Regional constrained controllability problem: Approaches and simulations

Hassan

Fatima

2009

Int. J. Control Autom. Syst.

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The aim of this paper is to deal with an output controllability problem. It consists in driving the state of a distributed parabolic system toward a state between two prescribed functions on a boundary subregion of the system evolution domain with minimum energy control. Two necessary conditions are derived. The first one is formulated in terms of subdifferential associated with a minimized functional. The second one is formulated as a system of equations for arguments of the Lagrange systems. Numerical illustrations show the efficiency of the second approach and lead to some conjectures.

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Applied Functional Analysis and Partial Differential Equations

Cited by 36 publications

References 0 publications

Balanced POD for model reduction of linear PDE systems: convergence theory

Balanced POD for model reduction of linear PDE systems: convergence theory

On the stability and error structure of BDF schemes applied to linear parabolic evolution equations

Regional constrained controllability problem: Approaches and simulations

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