A semigroup theoretic approach to modeling of boundary input problems

Washburn, Donald C.

doi:10.1007/bfb0003753

Cited by 10 publications

(6 citation statements)

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“…Simultaneously, and in parallel fashion, the aforementioned investigative efforts since the mid 1970s also produced "abstract models" for mixed PDE problems subject to control either acting on the boundary of, or else as a point control within, a multidimensional bounded domain, see [2,82,83] for parabolic problems and [24,25,73] for hyperbolic problems. Though, in particular, operators arising in the abstract model depend on both the specific class of PDEs and its specific homogeneous and nonhomogeneous boundary conditions, one cardinal point reached in this line of investigation was the following discovery: most of them-but by no means all of them [9,23, 78]-are encompassed and captured by the abstract model…”

Section: Lasiecka and R Triggiani 1063mentioning

confidence: 99%

L₂(Σ)‐regularity of the boundary to boundary operator B^∗L for hyperbolic and Petrowski PDEs

Lasiecka

Triggiani

2003

Abstract and Applied Analysis

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This paper takes up and thoroughly analyzes a technical mathematical issue in PDE theory, while-as a by-pass product-making a larger case. The technical issue is the L 2 (Σ)-regularity of the boundary → boundary operator B * L for (multidimensional) hyperbolic and Petrowski-type mixed PDEs problems, where L is the boundary input → interior solution operator and B is the control operator from the boundary. Both positive and negative classes of distinctive PDE illustrations are exhibited and proved. The larger case to be made is that hard analysis PDE energy methods are the tools of the trade-not soft analysis methods. This holds true not only to analyze B * L but also to establish three inter-related cardinal results: optimal PDE regularity, exact controllability, and uniform stabilization. Thus, the paper takes a critical view on a spate of "abstract" results in "infinite-dimensional systems theory," generated by unnecessarily complicated and highly limited "soft" methods, with no apparent awareness of the high degree of restriction of the abstract assumptions made-far from necessary-as well as on how to verify them in the case of multidimensional dynamical systems such as PDEs.1. A historical overview: hard analysis beats soft analysis on regularity, exact controllability, and uniform stabilization of hyperbolic and Petrowski-type PDEs under boundary control At first, naturally, PDEs boundary control theory for evolution equations tackled the most established PDE classes-parabolic PDEs-whose Hilbert space theory for mixed problems was already available in a form close to an optimal book form [51,56,57,58] since the early 1970s. Next, in the early 1980s, when the study of boundary control problems for (linear) PDEs began to address hyperbolic and Petrowski-type systems on a multidimensional bounded domain [10,26] (see [5,6, 35,44,45] Hard analysis energy methods. A happy and quite challenging exception was the optimal-both interior and boundary-regularity theory for mixed, nonsymmetric, noncharacteristic first-order hyperbolic systems culminated through repeated efforts in the early 1970s [16,63,64]. Its final, full success required eventually the use of pseudodifferential energy methods (Kreiss' symmetrizer). Apart from this isolated case, mathematical knowledge of global optimal regularity theory of hyperbolic and Petrowski-type mixed problems was scarce, save for some trivial one-dimensional cases. Thus, in this gloomy scenario, one may say that optimal control theory [10, 26, 51] provided a forceful impetus in seeking to attain an optimal global regularity theory for these classes of mixed PDEs problems. To this end, PDEs (hard analysis) energy methods-both in differential and pseudodifferential form-were introduced and brought to bear on these problems. The case of second-order hyperbolic equations under Dirichlet boundary control was tackled first. The resulting theory that emerged turns out to be optimal and does not depend on the space dimension [22,24,25,43,52]. It was best achieved by the use of energ...

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Section: Lasiecka and R Triggiani 1063mentioning

confidence: 99%

L₂(Σ)‐regularity of the boundary to boundary operator B^∗L for hyperbolic and Petrowski PDEs

Lasiecka

Triggiani

2003

Abstract and Applied Analysis

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“…It is a pleasure to acknowledge that our present work is inspired, in its conception, by the recent formulation of Balakrishnan and Washburn [3,4], [34,35] aimed at modeling, within the semigroup theory framework, nonsmooth Dirichlet boundary value problems for second-order parabolic equations. Their analysis, however, relies heavily on the assumption, characteristic of diffusion processes, that the semigroup be analytic.…”

Section: Uehi([ot]®f) Foehl(s2)mentioning

confidence: 98%

A cosine operator approach to modelingL 2(0,T; L 2 (?))?Boundary input hyperbolic equations

1981

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Abstract. This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain fl with boundary F, under the action of a boundary forcing term in L2(0, T; L2(F)). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is in L2(0 , T; L2(9)), when 9 is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is in L2(0, T; 3 n~-(~)) when 9 is a parallelepiped and in L2 (0, T; Ha (9)when ~ is a sphere. The Dirichlet case for general domains is Studied by means of pseudodifferential operator techniques.

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“…Simultaneously, and in parallel fashion, the aforementioned investigative efforts since the mid-70's also produced "abstract models" for mixed PDE problems, subject to control either acting on the boundary of, or else as a point control, within a multidimensional bounded domain: [2,93,94] for parabolic problems; [80,26,27] for hyperbolic problems. Though, in particular, operators arising in the abstract model depend on both the specific class of PDEs and on its specific homogeneous and nonhomogeneous boundary conditions, one cardinal point reached in this line of investigation was the following discovery: that most of them-but by no means all [7,23,86]-are encompassed and captured by the abstract model:…”

Section: Abstract Models Of Pde Mixed Problemsmentioning

confidence: 99%

Sobolev Spaces in Mathematics III

Isakov¹

2009

International Mathematical Series

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A semigroup theoretic approach to modeling of boundary input problems

Cited by 10 publications

References 3 publications

L₂(Σ)‐regularity of the boundary to boundary operator B^∗L for hyperbolic and Petrowski PDEs

L₂(Σ)‐regularity of the boundary to boundary operator B^∗L for hyperbolic and Petrowski PDEs

A cosine operator approach to modelingL 2(0,T; L 2 (?))?Boundary input hyperbolic equations

Sobolev Spaces in Mathematics III

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A semigroup theoretic approach to modeling of boundary input problems

Cited by 10 publications

References 3 publications

L2(Σ)‐regularity of the boundary to boundary operator B∗L for hyperbolic and Petrowski PDEs

L2(Σ)‐regularity of the boundary to boundary operator B∗L for hyperbolic and Petrowski PDEs

A cosine operator approach to modelingL 2(0,T; L 2 (?))?Boundary input hyperbolic equations

Sobolev Spaces in Mathematics III

Contact Info

Product

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L₂(Σ)‐regularity of the boundary to boundary operator B^∗L for hyperbolic and Petrowski PDEs

L₂(Σ)‐regularity of the boundary to boundary operator B^∗L for hyperbolic and Petrowski PDEs