# A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

Alexander Mielke; Michael Ortiz

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 14, Issue: 3, page 494-516
- ISSN: 1292-8119

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topMielke, Alexander, and Ortiz, Michael. "A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 494-516. <http://eudml.org/doc/90880>.

@article{Mielke2007,

abstract = {
This work is concerned with the reformulation of evolutionary problems in a
weak form enabling consideration of solutions that may exhibit
evolving microstructures. This reformulation is accomplished by
expressing the evolutionary problem in variational form, i.e., by
identifying a functional whose minimizers represent entire
trajectories of the system. The particular class of functionals under
consideration is derived by first defining a sequence of time-discretized
minimum problems and subsequently formally passing to the limit of
continuous time. The resulting functionals may be regarded as a
weighted dissipation-energy functional with a weight decaying with a
rate $1/\epsilon$. The corresponding Euler-Lagrange equation leads to an
elliptic regularization of the original evolutionary problem. The
Γ-limit of these functionals for $\epsilon\to 0$ is highly
degenerate and provides limited
information regarding the limiting trajectories of the system. Instead we
seek to characterize the minimizing trajectories directly. The special class
of problems characterized by a rate-independent dissipation
functional is amenable to a particularly illuminating analysis. For these
systems it is possible to derive a priori bounds that are independent
of the regularizing parameter, whence it is possible to extract convergent
subsequences and find the limiting trajectories. Under general assumptions
on the functionals, we show that all such limits satisfy the energetic
formulation (S) & (E) for rate-independent systems. Moreover, we show
that the accumulation points of the regularized solutions solve the
associated limiting energetic formulation.
},

author = {Mielke, Alexander, Ortiz, Michael},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Weighted energy-dissipation functional; incremental minimization problems; relaxation of evolutionary problems; rate-independent processes; energetic solutions; weighted energy-dissipation functional},

language = {eng},

month = {12},

number = {3},

pages = {494-516},

publisher = {EDP Sciences},

title = {A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems},

url = {http://eudml.org/doc/90880},

volume = {14},

year = {2007},

}

TY - JOUR

AU - Mielke, Alexander

AU - Ortiz, Michael

TI - A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/12//

PB - EDP Sciences

VL - 14

IS - 3

SP - 494

EP - 516

AB -
This work is concerned with the reformulation of evolutionary problems in a
weak form enabling consideration of solutions that may exhibit
evolving microstructures. This reformulation is accomplished by
expressing the evolutionary problem in variational form, i.e., by
identifying a functional whose minimizers represent entire
trajectories of the system. The particular class of functionals under
consideration is derived by first defining a sequence of time-discretized
minimum problems and subsequently formally passing to the limit of
continuous time. The resulting functionals may be regarded as a
weighted dissipation-energy functional with a weight decaying with a
rate $1/\epsilon$. The corresponding Euler-Lagrange equation leads to an
elliptic regularization of the original evolutionary problem. The
Γ-limit of these functionals for $\epsilon\to 0$ is highly
degenerate and provides limited
information regarding the limiting trajectories of the system. Instead we
seek to characterize the minimizing trajectories directly. The special class
of problems characterized by a rate-independent dissipation
functional is amenable to a particularly illuminating analysis. For these
systems it is possible to derive a priori bounds that are independent
of the regularizing parameter, whence it is possible to extract convergent
subsequences and find the limiting trajectories. Under general assumptions
on the functionals, we show that all such limits satisfy the energetic
formulation (S) & (E) for rate-independent systems. Moreover, we show
that the accumulation points of the regularized solutions solve the
associated limiting energetic formulation.

LA - eng

KW - Weighted energy-dissipation functional; incremental minimization problems; relaxation of evolutionary problems; rate-independent processes; energetic solutions; weighted energy-dissipation functional

UR - http://eudml.org/doc/90880

ER -

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