Volume 7 (2001), No. 1 (January), pp. 1--149

The main idea of this paper is to apply a dynamic improvement technique to linear constrained control continuous-time systems with a state observer. Two known cases with the full-order observer and the minimal-order observer are used for reconstructing the inaccessible state. This leads to using a large set of initial states where the control saturation does not occur to the cost of a slow dynamic of the closed-loop. As soon as the state tends to the origin, the dynamics becomes better and better while the constraints on the control vector are respected.

S.Kh. Aranson and E.V. Zhuzhoma. On the C^r-Closing Lemma and the Koebe--Morse Coding of Geodesics on Surfaces. J. Dynam. Control Systems 7 (2001), no. 1, 49--59.

We consider the C^r-closing lemma (r \geq 2) for vector fields with finitely many singularities on an orientable closed surface M of genus g \geq 2. Given a nontrivially recurrent trajectory, there is the corresponding geodesic having the same asymptotic directions (both negative and positive). Using the Koebe--Morse coding for the corresponding geodesic, we introduce the notion of p-expansions in the form of two sequences of nonnegative integers. The main result is the following. Suppose a vector field X \in \chi ^r(M), r \geq 2, has a nontrivially recurrent trajectory l through a point m; then there exists Y \in \chi ^r(M) arbitrarily close to X (in the C^r-topology) having a periodic trajectory through m provided that the Koebe--Morse coding of the corresponding geodesic g(l) has p-expansions of unrestricted type.

F. Colonius and Du Weihua. Hyperbolic Control Sets and Chain Control Sets. J. Dynam. Control Systems 7 (2001), no. 1, 49--59.

A shadowing lemma for hyperbolic control flows is proved. A consequence is that hyperbolic chain control sets are closures of control sets.

Dirk Mittenhuber. Controllability of Systems on Solvable Lie Groups: the Generic Case. J. Dynam. Control Systems 7 (2001), no. 1, 61--75.

A Lie group G with Lie algebra g is called SID-controllable if there exist A, B in g such that the (Single Input with Drift) control system dg/dt=g(A+uB), u \in R, is controllable. This is equivalent to saying that the semigroup S(A,\pm B) generated by exp(\R^+ A) \cup exp(R B) is all of G.

This definition is due to Sachkov who also classified SID-controllable solvable Lie algebras. It turns out that SID-controllability is actually a property of the Lie algebra (rather than of a control system): if a solvable g is SID-controllable, then a generic SID-system will be controllable.

In this paper we generalize this result to systems with multiple inputs and drifts: G is I_nD_m-controllable if there exist ``inputs'' B_1 ... B_n and ``drifts'' A_1 ... A_m such that S(A_1, ... ,A_m; \pm B_1, ... ,\pm B_n)=G. A Lie algebra is called I_nD_m-controllable if the corresponding simply connected group has this property.

We will show that every solvable Lie algebra has a generic controllability rank r \in N and a generic controllability type (i,d) \in N \times N_0 such that:

(GCR)
g is not I_{r-1}-controllable, g is I_r-controllable, and the latter is generic. In particular, g is I_nD_m-controllable if n \geq r or n+m>r;

(GCT)
g is I_iD_d-controllable, a generic I_iD_d-system is controllable, i+d=r, and i is minimal.

Determination of these invariants is one of our goals. Our major tools will be reduction arguments which are of independent interest: we show that every solvable Lie algebra g has a maximal ideal i which is completely irrelevant for all controllability questions. Passing to the factoralgebra g/i and analyzing its structure is the key step in solving our problem.

A.A. Davydov and J. Basto-Concalves. Controllability of Generic Inequalities Near Singular Points. J. Dynam. Control Systems 7 (2001), no. 1, 77--99.

In this paper the authors obtained some sufficient conditions under which one has the small local controllability property for a general nonlinear control system described by a system of nonlinear inequalities.

M. Canalis-Durand, F. Michel, and M. Teisseyre. Algorithms for Formal Reduction of Vector Field Singularities. J. Dynam. Control Systems 7 (2001), no. 1, 101--125.

We present several algorithms transforming a Pfaffian form of the following type:
\omega =d(y^2-x^q)+\Delta (x,y)(2x +dy-qy dx)
into the normal form given by F. Loray. We are interested only in the cases q=2 and q=3, where the latter represents the generic case of odd q. Here \Delta (x,y) is a convergent series which does not contain a constant term. Our results give a numerical answer to the question on the convergence of normal forms. For q=3, in the nondegenerate case, our numerical analysis of algorithms leads to Gevrey estimates of order 2. Thus, our proposed models are presumably not analytic. Moreover, in the case q=2, our results agree with the results of Martinet and Ramis.

A.Ya. Kazakov. Coalescence of Two Regular Singularities into One Regular Singularity for the Linear Ordinary Differential Equation. J. Dynam. Control Systems 7 (2001), no. 1, 127--149.

The process of coalescence of two regular points of a linear ODE into a single regular singularity is studied. This problem is examined in terms of connection matrices and solutions of the corresponding equations.