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1

A continuum description of failure waves

Said H., Glimm J.

Journal of Theoretical and Applied Mechanics

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2018

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Vol. 56 nr 3

657—674

EN

Shattering of a brittle material such as glass occurs dynamically through a propagating failure wave, which however, can not be assigned to any of the classical wave. In this paper, we build a thermodynamically consistent theory based on the idea that a failure wave is analogous to a deﬂagration wave. Our theory admits, as special cases, the classical models of Feng and Clifton. Two fundamental thermodynamic functions form the basis of our theory. This approach allows for the construction of a new variational principle and a Lagrangian formulation. Finally, the theory is linearized to interpret speciﬁc experimental observations.

2

Attractors of strongly dissipative systems

Ramm A. G.

Bulletin of the Polish Academy of Sciences. Mathematics

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2009

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Vol. 57, no 1

25-31

EN

A class of infinite-dimensional dissipative dynamical systems is denned for which there exists a unique equilibrium point, and the rate of convergence to this point of the trajectories of a dynamical system from the above class is exponential. All the trajectories of the system converge to this point as t → +∞, no matter what the initial conditions are. This class consists of strongly dissipative systems. An example of such systems is provided by passive systems in network theory (see, e.g., MR0601947 (83m:45002)).

3

J-energy preserving well-posed linear systems

Staffans O. J.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 6

1361-1378

EN

The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE's, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foias. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H2- and Hinfty-theories).

4

Factorization of the Popov Function of a Multivariable Linear Distributed Parameter System in the Non-Coercive Case: a Penalization Approach

Pandolfi L.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 6

1249-1260

EN

We study the construction of an outer factor to a positive definite Popov function Pi(iomega) of a distributed parameter system. We assume that Pi(iomega) is a non-negative definite matrix with non-zero determinant. Coercivity is not assumed. We present a penalization approach which gives an outer factor just in the case when there exists any outer factor.