iISS and ISS dissipation inequalities: preservation and interconnection by scaling

2016 ◽  
Vol 28 (3) ◽  
Author(s):  
Hiroshi Ito ◽  
Christopher M. Kellett
Keyword(s):  
Dissipation Inequalities

scholarly journals On dissipation inequalities and linear viscoelasticity

1965 ◽  
Vol 23 (3) ◽  
pp. 235-245 ◽  
Author(s):  
M. E. Gurtin ◽  
I. Herrera
Keyword(s):  
Linear Viscoelasticity ◽  
Dissipation Inequalities

scholarly journals THERMODYNAMICALLY CONSISTENT, FRAME INDIFFERENT DIFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE TWO-PHASE FLOWS WITH DIFFERENT DENSITIES

2012 ◽  
Vol 22 (03) ◽  
pp. 1150013 ◽  
Author(s):  
HELMUT ABELS ◽  
HARALD GARCKE ◽  
GÜNTHER GRÜN
Keyword(s):  
Surface Diffusion ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Momentum Balance ◽  
Two Phase ◽  
Interface Models ◽  
Natural Energy ◽  
Soluble Species ◽  
Sharp Interface Models ◽  
Dissipation Inequalities

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation, we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

scholarly journals ISS and integral-ISS of switched systems with nonlinear supply functions

2021 ◽  
Author(s):  
Shenyu Liu ◽  
Aneel Tanwani ◽  
Daniel Liberzon
Keyword(s):  
Switched Systems ◽  
Lyapunov Functions ◽  
Dwell Time ◽  
Average Dwell Time ◽  
Switched Nonlinear Systems ◽  
Activation Time ◽  
Unstable Subsystems ◽  
Supply Functions ◽  
Dissipation Inequalities ◽  
Integral Version

AbstractThe problem of input-to-state stability (ISS) and its integral version (iISS) are considered for switched nonlinear systems with inputs, resets and possibly unstable subsystems. For the dissipation inequalities associated with the Lyapunov function of each subsystem, it is assumed that the supply functions, which characterize the decay rate and ISS/iISS gains of the subsystems, are nonlinear. The change in the value of Lyapunov functions at switching instants is described by a sum of growth and gain functions, which are also nonlinear. Using the notion of average dwell-time (ADT) to limit the number of switching instants on an interval, and the notion of average activation time (AAT) to limit the activation time for unstable systems, a formula relating ADT and AAT is derived to guarantee ISS/iISS of the switched system. Case studies of switched systems with saturating dynamics and switched bilinear systems are included for illustration of the results.

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