It is well known that Kirchhoff equations admit infinitely many simple modes, i. e. time periodic solutions with only one Fourier component in the space variables. We prove that, for some choices of the nonlinearity, these simple modes are unstable provided that their energy is large enough. This result, stated in an abstract Hilbert space setting, and proved by reducing to a system of two seco...

In this article, we investigate the exact controllability of the 2DSchrödinger-Poisson system, which couples a Schrödinger equation on a bounded domain of R with a Poisson equation for the electrical potential. The control acts on the system through a Neumann boundary condition on the potential, locally distributed on the boundary of the space domain. We prove several results, with or without n...

We derive decay rates for the energy of solutions of dissipative wave equations. The metod of proof combines multiplier techniques and the construction of suitable Lyapunov functionals. Without imposing any growth condition at the origin on the nonlinearity we show that this Lyapunov functional, which is equivalent to the energy of the system, is bounded above by the solution of a differential ...

Starting from the equations occurring in flight control, there are considered time delay systems with (possibly) discontinuous but bounded nonlinearities. A useful definition of the solution is based on the so-called extended nonlinearity which allows development of the standard absolute stability theory; within this theory a stability result is extended to a significant class of time delay sys...

In almost all papers in the literature, the results on exact controllability hold only for finite dimensional Banach spaces, since compactness of the semigroup and the bounded invertibility of an operator implies finite dimensional. In this note we show that the existence theory on controllability in the literature, can trivially be adjusted to include the infinite dimensional space setting, if...

We show that there is a convex ring R = Ω \Ω ⊂ R in which there exists a solution u to a semilinear partial differential equation ∆u = f(u), u = −1 on ∂Ω, u = 1 on ∂Ω, with level sets, not all convex. Moreover every bounded solution u has at least one nonconvex level set. In our construction, the nonlinearity f , is non-positive, and smooth. AMS Classification: 35J60, 35R35.

This paper presents a technique for designing switching rules that drive the state of a class of nonlinear switched system to a desired constant reference. The system may contain state-dependent sector-bounded nonlinear functions. The proposed method considers a switching rule using the ‘max’ composition of auxiliary functions. The results are given in terms of Linear Matrix Inequalities (LMIs)...

We study the qualitative properties of the generalized transition fronts for the reactiondiffusion equations with the spatially inhomogeneous nonlinearity of the ignition type. We show that transition fronts are unique up to translation in time and are globally exponentially stable for the solutions of the Cauchy problem. The results hold for reaction rates that have arbitrary spatial variation...

In this paper we use the method of upper and lower solutions combined with degree theoretic techniques to prove the existence of multiple positive solutions to semipositone superlinear systems of the form −∆u = g1(x, u, v) −∆v = g2(x, u, v) on a smooth, bounded domain Ω ⊂ R with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical,...