Adaptive stabilization with exponential decay

Domination with exponential decay

Let G be a graph and S ⊆ V (G). For each vertex u ∈ S and for each v ∈ V (G) − S, we define d(u, v) = d(v, u) to be the length of a shortest path in 〈V (G)−(S−{u})〉 if such a path exists, and∞ otherwise. Let v ∈ V (G). We define wS(v) = ∑ u∈S 1 2d(u,v)−1 if v 6∈ S, and wS(v) = 2 if v ∈ S. If, for each v ∈ V (G), we have wS(v) ≥ 1, then S is an exponential dominating set. The smallest cardinalit...

Stabilization of Oscillators Subject to Dry Friction: Finite Time Convergence Versus Exponential Decay Results

We investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion (S) ẍ(t) + ∂Φ(ẋ(t)) + ∇f(x(t)) 3 0, t ≥ 0, where f : R → R is a smooth potential and Φ : R → R is a convex function. The friction is modelized by the subdifferential term −∂Φ(ẋ). When 0 ∈ int(∂Φ(0)) (dry friction condition), it is shown in [1] that the unique solution of (S) converges i...

Exponential Decay in Nonlinear Networks1

Almost Exponential Decay near Maxwellian

By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in [7–9, 11], we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for char...

Is Radioactive Decay Really Exponential?

Radioactive decay of an unstable isotope is widely believed to be exponential. This view is supported by experiments on rapidly decaying isotopes but is more difficult to verify for slowly decaying isotopes. The decay of 14C can be calibrated over a period of 12,550 years by comparing radiocarbon dates with dates obtained from dendrochronology. It is well known that this approach shows that rad...