Volume 6 (2000), No. 2 (April), pp. 159--309

Yu.L. Sachkov. Classification of Controllable Systems on Low-Dimensional Solvable Lie Groups. J. Dynam. Control Systems 6 (2000), no. 2, 159--217.

Right-invariant control systems on simply connected solvable Lie groups are studied. A complete and explicit description of controllable single-input right-invariant systems on such Lie groups up to dimension 6 is obtained.

A.I. Gladyshev. On the Riemann--Hilbert Problem in Dimension 4. J. Dynam. Control Systems 6 (2000), no. 2, 219--264.

We consider the problem of existence of a Fuchsian system with a prescribed 4-dimensional monodromy. We give a classification of all cases of negative solution of this problem in terms of reducibility pattern of the representation, its local structure (which is described by a modification of Jordan form), and restrictions on asymptotics of solutions to Fuchsian systems in lower dimensions. We also show that realization of a reducible 4-dimensional representation by a Fucshian system, if it exists, can be chosen in a block upper-triangular form (though not necessarily with the same reducibility pattern). At the end of the paper, we present new counterexamples to the Riemann--Hilbert problem in dimension 4.

M. Guerra. J. Dynam. Control Systems 6 (2000), no. 2, 265--309.

It is well known that singular problems may fail to have optimal solution in the class of ``ordinary'' (say, square-integrable) controls, even in the cases where the cost is bounded from below. In this paper, we suggest a method for overcoming this difficulty by defining an order r of singularity of the problem and extending both the input-trajectory map and the cost functional to an adequate subspace of the Sobolev space H_{-r}. We show that the extended problem has a minimum if and only if the infimum of the original problem is finite. The extended problem can be transformed in a ``natural'' way into a regular L-Q problem with strictly smaller controllable space and (possibly) smaller control space. We use this transformation to describe the structure of relaxed optimal controls and the corresponding relaxed trajectories. We provide a method for computing optimal relaxed solutions from the solution of an adequate Riccati differential equation. We also show how the relaxed minimizers can be approximated by square-integrable functions.