Boundary control of weakly nonlinear wave equations with an application to galloping of transmission lines

Abstract

Abstract. Exponential decay is proven for a class of initial boundary value problems for the equation w u -CoWx~ = f (wt). The boundary condition is CoW x + rw t = 0 at x = 1 and w(0, t) = 0. Iffsatisfies a global Lipschitz condition, no restrictions are placed on the initial conditions, but if this is relaxed to a local Lipschitz condition, the initial data are assumed to be sufficiently small. These theorems are motivated in part by an application to modeling of "galloping" transmission lines. A theorem about boundedness of solutions without boundary damping is proven also. A global Lipschitz condition is assumed here, but the theorem is believed to be more generally true.

Global dynamics of nonlinear wave equations with cubic non-monotone damping

“…The equation (2.1) in one spatial dimension (n = 1) with f (s) = 0 and g(s) given in (2.2) has been exploited as a mathematical model of galloping vibrations of power lines in distant electric transmission, cf. [12] and [13]. The corresponding results on the existence of global solutions and dissipative dynamics properties have been proved in [14].…”

“…The corresponding results on the existence of global solutions and dissipative dynamics properties have been proved in [14]. We noticed from [12] and [13] that the primary concern from the engineering viewpoint is whether for any initial status with finite energy a solution exists globally for t > 0 without blow-up, and whether every global solution remains to be bounded in the energy space E, here E = V × H.…”