Distributed Systems with Persistent Memory

Pandolfi, Luciano

doi:10.1007/978-3-319-12247-2

Cited by 26 publications

(60 citation statements)

References 63 publications

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“…The fact that f ↦ w is a linear continuous transformation from L 2 ( ∂ Ω × (0, T 0 )) to C ([0, T 0 ]; L 2 (Ω)) implies that the moment operator

scriptM

is continuous from L 2 ( ∂ Ω × (0, T 0 )) to l 2 , and controllability shows that it is surjective. So the sequence of functions

\{γ_{1} ϕ_{n} \int_{0}^{s} N (s - r) z_{n} (r) d r\}

is a Riesz sequence in L 2 ( ∂ Ω × (0, T 0 )), and this in turn implies that the series

\sum_{n = 1}^{+ \infty} ξ_{n} [γ_{1} ϕ_{n} \int_{0}^{s} N (s - r) z_{n} (r) d r]

converges in L 2 ( ∂ Ω × (0, T )) for every T (both larger and and less than T 0 ) and for every { ξ n }∈ l 2 .…”

Section: Appendix: the Series (7)mentioning

confidence: 88%

“…We use

f (x, t) = f_{0} (\overset{x}{true}) f_{1} (t)

in (), and we define

f_{0, n} = [\int_{\hat{Γ}} γ_{1} ϕ_{n} (\hat{x}) f_{0} (\hat{x}) d \hat{Γ}] .

Then we have

w_{n}^{'} (t) = - λ_{n}^{2} \int_{0}^{t} N (t - s) w_{n} (s) d s - f_{0, n} \int_{0}^{t} N (t - s) f_{1} (s) d s

so that (we use f 1 (0) = 0 and the variation of constants formula; see )

\begin{matrix} w_{n} (t) & = - f_{0, n} {falsefalse}_{\int}^{0} z_{n} (t - r) {falsefalse}_{\int}^{0} N (r - s) f_{1} (s) normald s normald r = - f_{0, n} {falsefalse}_{\int}^{0} f_{1} (s) {falsefalse}_{\int}^{0} N (t - s - r) z<...> \end{matrix}

…”

Section: Justification Of the Algorithmmentioning

confidence: 99%

“…We need few information. A sequence { e n } in a (real or complex) Hilbert space is a Riesz sequence if it can be transformed to an orthonormal basis (possibly of a Hilbert space of different dimension) using a linear bounded and boundedly invertible transformation defined on cl span{ e n } (see for Riesz sequences). It is a fact that when { e n } is a Riesz sequence, then the series

\sum_{n = 1}^{+ \infty} h_{n} e_{n}

converges if and only if { h n }∈ l 2 and the sum does not depend on the order of the elements of the series.…”

Section: Appendix: the Series (7)mentioning

confidence: 99%

“…The following PDE with persistent memory

w' = {falsefalse}_{\int}^{0} N (t - s) normalΔ w (s) normald s

(Δ is the Laplacian, and the apex denotes time derivative, w = w ( t ) = w ( x , t ), with x ∈Ω—the variable x is not indicated unless needed for clarity) is encountered in the study of diffusion processes in the presence of complex molecular structures ( w = w ( x , t ) is the concentration; see ) or in thermodynamics of materials with memory ( w is the temperature; see ). It is also encountered in viscoelasticity, usually written as follows:

w^{''} = N (0) w (x, t) + \int_{0}^{t} N^{'} (t - s) Δ w (s) d s,

(here, w is the displacement; see ).…”

Section: Introductionmentioning

confidence: 99%

“…These assumptions in particular imply the following: System () with conditions w ( x ,0) = ξ ( x ) , w ( x , t ) = f ( x , t ) x ∈ ∂ Ω is solvable for every ξ ∈ L 2 (Ω) and f ∈ L 2 ( ∂ Ω × (0, T )), and it turns out that w ∈ C ([0, T ]; L 2 (Ω)) × C 1 ([0, T ]; H − 1 (Ω)). Variations of the temperature propagate with finite speed

\sqrt{N (0)}

(in every direction, like in the wave equation, see ). System () is controllable in the following sense: There exists a time T 0 such that for every η ∈ L 2 (Ω), there exists f ∈ L 2 ( ∂ Ω × (0, T 0 )) such that w ( T 0 ) = η (the initial condition is zero, and we assume w = f on ∂ Ω). See for the proof which uses both the regularity of the kernel, and the condition N (0) ≠ 0. …”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution

Pandolfi

2016

Math Methods in App Sciences

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We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body. The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.

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“…The fact that f ↦ w is a linear continuous transformation from L 2 ( ∂ Ω × (0, T 0 )) to C ([0, T 0 ]; L 2 (Ω)) implies that the moment operator