A New Approach to Equations with Memory

By Mauro Fabrizio, Claudio Giorgi, Vittorino Pata. 2009

156 citations recorded by the source corpus.

AbstractWe discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function $${G : \mathbb{R}^+ \to \mathbb{R}^+}$$ such that $$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $$we consider an abstract version of the evolution equation $$\partial_{tt}{\varvec u}({\varvec x}, t) - \Delta\left[G(0){\varvec u}({\varvec x}, t) + \displaystyle\int_0^\infty G'(s){\varvec u}({\varvec x}, t - s){\rm d}s\right] = 0$$arising from linear viscoelasticity.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/9a5c49c70dda48390fbd84afbf399027699cad34

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Vittorino Pata
2. Monica Conti
3. Bernard Helffer
4. S. Terracini
5. Gianmaria Verzini
6. Igor D. Chueshov

In Control and Systems Engineering

1. Vittorino Pata
2. Claudio Giorgi
3. Monica Conti
4. Valeria Danese
5. Aissa Guesmia
6. To Fu Ma
7. Igor D. Chueshov

Citations in the corpus

• A model of viscoelasticity with time-dependent memory kernels (2016, background): “It is worth mentioning also the possibility of extending in a quite natural way the underlying ideas to the study of systems with memory in the so-called minimal state framework introduced in [15].”