Volume 6 (2000), No. 4 (October), pp. 461--602

H. Marzougui. Flows with infinite singularities on closed two-manifolds. J. Dynam. Control Systems 6 (2000), no. 4, 461--476.

We consider flows with infinitely many singularities on closed orientable 2-manifolds. We give a description of the flow near a leaf and apply this result to transverse invariant measures.

E. Litsyn, Yu.V. Nepomnyashchikh, A. Ponosov. Classification of linear dynamical systems in the plane admitting a stabilizing hybrid feedback control. J. Dynam. Control Systems 6 (2000), no. 4, 477--501.

We suggest a classification of linear controlled planar systems and describe those of them which admit and which do not admit a stabilization via the so-called ``linear hybrid feedback controls.'' We give also an explicit description of all the parameters of the stabilizing control procedure in terms of certain inequalities.

S.A. Agafonov. On the stability of nonconservative systems with estimation of the attraction domain. J. Dynam. Control Systems 6 (2000), no. 4, 503--510.

Mechanical systems subjected to dissipative, gyroscopic, conservative, and also nonconservative positional forces are considered.

The question of the effect of dissipative, gyroscopic, and conservative forces on the motion stability of a mechanical systems is determined by classical Kelvin--Chetaev theorems [1]. The presence of nonconservative positional forces considerably complicates the situation and excludes direct application of these theorems. Applying Lyapunov's functions method the condition of asymptotic stability of a mechanical system under the action of all listed above forces is obtained. Moreover, the estimation of the attraction domain in phase space is found. The precession system which is used in the solution of some problems in the applied theory of the gyroscopic systems is also examined. The connection between the stability of origin and precession systems is detected.

Theoretical results are applied to the stabilization problem of stationary motion of the balanced gimbal suspension gyro by means of external moments.

E. Trelat. Some properties of the value function and its level sets for affine control systems with quadratic cost. J. Dynam. Control Systems 6 (2000), no. 4, 511--541.

Let T>0 be fixed. We consider the optimal control problem for analytic affine systems: \dot{x}=f_0(x) + \sum_{i=1}^m u_i f_i(x), with a cost of the form: C(u)=\int_0^T \sum_{i=1}^m u_i^2(t) dt. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function S is subanalytic. Second, we prove that if there exists an abnormal minimizer of corank 1, then the set of endpoints of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.

A. Benabdallah, A. Soufyane. Uniform stability and stabilization of linear thermoelastic systems. J. Dynam. Control Systems 6 (2000), no. 4, 543--560.

The object of this work is the study of uniform stability and stabilization of linear thermoelastic systems. We construct an explicit stabilizer feedback for the thermoelastic system and use a Lyapunov function to prove the exponential stability.

F. Dal'bo, A.N. Starkov. On a classification of limit points of infinitely generated Schottky groups. J. Dynam. Control Systems 6 (2000), no. 4, 561--578.

Let \L\subset\dH^2 be the limit set of a Fuchsian group \G\subset \Iso\H^2. The behavior of geodesic and horocycle trajectories on the unit tangent bundle \G\bs T^1\H^2 allows one to introduce different subclasses of \L such as the conical limit set \L_c\subset\L, the horocyclic limit set \L_h\subset\L, etc. We consider a class of infinitely generated Schottky groups obtained by placing countably many disjoint semicircles in \H^2 with a single accumulation point in \dH^2, and pairing them by hyperbolic transformations. For all such groups we give a complete description of \L_c\subset\dH^2 and of all other subclasses of \L related to the dynamics of the geodesic flow on \G\bs T^1\H^2. This is given in terms of natural combinatorial coding of limit points and is the same for all groups \G of this form.

In contrast, we show that the structure of horocycle orbits is very sensitive to geometric parameters of the semicircles nearby the accumulation point. In particular, we construct an infinitely generated Schottky group such that the horocycle flow on T^1\H^2 has only two types of orbits: those dense in the nonwandering subset \Om^+ \subset T^1\H^2 and those divergent in both directions. This shows that infinitely generated Fuchsian group need not have Garnett limit points.

C. Bonatti, V. Grines. Knots as topological invariants for gradient-like diffeomorphisms of the sphere S^3. J. Dynam. Control Systems 6 (2000), no. 4, 579--602.

We show how knots in S^2\times S^1 appear in a natural way as complete invariants of topological conjugacy for the simplest gradient-like diffeomorphisms on 3-manifolds.