Volume 7 (2001), No. 3 (July), pp. 299--446

M. Boulanouar. Study of the Streaming Operator with General Maxwell's Boundary Conditions. J. Dynam. Control Systems 7 (2001), no. 3, 299--326.

We study the streaming operator in slab domain with general Maxwell's boundary conditions described by linear boundary operators K^+ and K^-. We prove that the streaming operator generates a strongly continuous semigroup. We show that the positivity and irreducibility of the boundary operators imply the positivity and irreducibility of the generated semigroup. We establish the spectral properties of the streaming operator and characterize its spectral bound. Under the compactness of boundary operators, we explicitly give the essential type of the generated semigroup and describe its asymptotic behavior in the operator norm topology.

G. Grammel. Exponential Stability via the Averaged System. J. Dynam. Control Systems 7 (2001), no. 3, 327--338.

An averaging theorem for nonlinear systems with bounded deterministic noise is proved in order to investigate the exponential stability of systems with fast periodic forcing. It turns out that uniform exponential stability of an equilibrium point of the averaged system is equivalent to robust uniform exponential stability of the perturbed system. Explicit estimates for both the approximation of single trajectories and the order of the exponential decay are obtained.

A. Agafonov. Gimbal Suspension Gyro: Stability, Bifurcation, and Chaos. J. Dynam. Control Systems 7 (2001), no. 3, 339--351.

In this paper the solutions of three problems, devoted to the investigation of the dynamics of balanced gimbal suspension gyro (GSG) which is installed at the immovable base are presented. In the first problem the stability of stationary motion (SM) of the GSG is analyzed under which the planes of the internal and external rings of GSG are orthogonal. The moment of viscous friction and the moment which is the function of the deviation angle of the internal ring act on the external axis. The analysis of stability of SM is carried out by means of construction of Lyapunov's function with the estimation of the attraction domain. In the second problem the mechanism of the stability loss of the SM under the transfer of the bifurcational parameter through the critical value is presented. In this case the periodic motion originates (Andronov--Hopf bifurcation). The orbital stability condition of the periodic motion is found. The third problem investigates the forced vibration of the GSG under the action on the internal ring of the perturbed moment which is the sum of the small moment of viscous friction and moment with small amplitude and fixed frequency. Here we consider the case where the projection of the angular moment on the axis of the external ring is equal to zero. In case of absence of the perturbations the SMs under which the external and internal rings are orthogonal or lie in the common plane are stable and unstable, respectively. For the unperturbed system the equation of the separatrix which passes through the hyperbolic points is found. For the determination of the condition of intersection of the separatrices in the perturbed system, the distance between them is calculated (Melnikov's distance). In the space of parameters the domain in which this distance can change the sign is distinguished and it is the feature of the chaotic motion arising.

Alejandro M. Meson and Fernando Vericat. Estimates for the Ghys--Langevin--Walczak Entropy. J. Dynam. Control Systems 7 (2001), no. 3, 353--365.

In this note, we present a calculation of the Ghys--Langevin--Walczak entropy (GLW-entropy) by using the growth rate of periodic points. We also introduce a ``functional'' entropy for dynamics given by operators on functional spaces. It serves as a bound for the GLW-entropy.

J. Bracho, L. Montejano, and D. Oliveros. A Classification Theorem for Zindler Carrousels. J. Dynam. Control Systems 7 (2001), no. 3, 367--384.

The purpose of this paper is to give a complete classification of Zindler carrousels with five chairs. This classification theorem gives enough evidence to show the nonexistence of figures different from the disk that float in equilibrium in every position for the corresponding perimetral densities.

Ugo Boscain and Benedetto Piccoli. Morse Properties for the Minimum Time Function on 2-D Manifolds. J. Dynam. Control Systems 7 (2001), no. 3, 385--423.

Given a two-dimensional smooth manifold M and two smooth vector fields X and Y on M, we want to steer a point p\in M to a point q\in M in minimum time using only integral curves of the vector fields X and Y. Fixing p, we define the minimum time function T_p(q) to reach q. We prove that, generically, T_p(q) is a Morse function in topological sense giving a positive answer to a question of V.I. Arnold.

Hubertus Th. Jongen and Oliver Stein. Nonconvex Optimization: Gradient Flows and Deformation. J. Dynam. Control Systems 7 (2001), no. 3, 425--446.

In this survey paper we consider two features in finite dimensional smooth nonconvex optimization problems. The first one is concerned with global aspects of gradient flows. The second part is devoted to one-parametric families of optimization problems. Here, the index set corresponding to inequality constraints might be infinite and might depend on the state variable, too.