Discrete and Continuous Dynamical Systems
January 2008 , Volume 20 , Issue 1
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Aim of this paper is to provide a survey of the theory of impulsive control of Lagrangian systems. It is assumed here that an external controller can determine the evolution of the system by directly prescribing the values of some of the coordinates. We begin by motivating the theory with a couple of elementary examples. Then we discuss the analytical form taken by the equations of motion, and their impulsive character. The following sections review various results found in the literature concerning the continuity of the control-to-trajectory map, the existence of optimal controls, and the asymptotic controllability to a reference state. In the last section we indicate a further application of the theory, to the control of deformable bodies immersed in a fluid.
For discrete competitive dynamical systems, amenable general conditions are presented to guarantee the existence of the carrying simplex and then these results are applied to age-structured semelparous population models, as well as to an annual plant competition model.
We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $\Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $\Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $L^\infty$ coefficients.
We consider linear nonautonomous second order parabolic equations on $\R^N$. Under an instability condition, we prove the existence of two complementary Floquet bundles, one spanned by a positive entire solution - the principal Floquet bundle, the other one consisting of sign-changing solutions. We establish an exponential separation between the two bundles, showing in particular that a class of sign-changing solutions are exponentially dominated by positive solutions.
We present some qualitative properties for solutions of singular quasilinear elliptic differential inequalities on complete Riemannian manifolds, such as the validity of the weak maximum principle at infinity, and non--existence results.
We discuss Fredholm properties of the linearization of partial differential equations on cylindrical domains about travelling and modulated waves. We show that the Fredholm index of each such linearization is given by a relative Morse index which depends only on the asymptotic coefficients. Several strategies are outlined that help to compute relative Morse indices using crossing numbers of spatial eigenvalues, and the results are applied to prove the existence of small viscous shock waves in hyperbolic conservation laws upon adding small localized time-periodic source terms.
The selecting lemma of Liao selects, under certain conditions of a non-hyperbolic setting, a special kind of orbits of finite length, called quasi-hyperbolic strings, which can be shadowed by true orbits. In this article we give an exposition on this lemma, and illustrate some recent applications.
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