R.V. Gamkrelidze. Discovery of the Maximum Principle. J. Dynam. Control Systems 5 (1999), no. 4, 437-- 451.
A short history of the discovery of the maximum principle in optimal control theory by L.S. Pontryagin and his associates is presented.
A.A. Bolibrukh. On Sufficient Conditions for the Existence of a Fuchsian Equation with Prescribed Monodromy. J. Dynam. Control Systems 5 (1999), no. 4, 453-- 472.
Recent sufficient conditions for positive solvability of the Riemann--Hilbert problem, known and new, are presented.
F. Monroy-Perez and A. Anzaldo-Meneses. Optimal Control on the Heisenberg Group. J. Dynam. Control Systems 5 (1999), no. 4, 473--499.
Let H denote either the Heisenberg group R^{2n+1}, or the
Cartesian product of n copies of the three-dimensional Heisenberg
group R^3. Let {X_1,Y_1, ... ,X_n,Y_n} be an independent
set of left-invariant vector fields on H. In this paper, we study
the left-invariant optimal control problem on H with the dynamics
\dot q(t)=\sum_{i=1}^n u_i(t) X_i(q(t))+v_i(t)Y(q(t)),
the cost functional
\Lambda(q,u)=1/2 \int \sum_{i=1}^n \mu_i(u_i^2+v_i^2),
with arbitrary positive parameters \mu_1, ... ,\mu_n, and
admissible controls taken from the set of measurable functions
t\mapsto u(t)=(u_1(t),v_1(t), ... ,u_n(t),v_n(t)).
The above control system is encoded either in the kernel of a contact
1-form (for R^{2n+1}), or in the kernel of a Pfaffian system (for
R^{3n}). In both cases, the action of the semi-direct product of the
torus T^n with H describe the symmetries of the problem.
The Pontryagin maximum principle provides optimal controls;
extremal trajectories are solutions to the Hamiltonian system
associated with the problem. Abnormal extremals (which do not
depend on the cost functional) yield solutions that are
geometrically irrelevant.
An explicit integration of the extremal equations provides a tool
for studying some aspects of the sub-Riemannian structure defined on H by
means of the above optimal control problem.
O.D. Anosova. On Invariant Manifolds in Singularly Perturbed Systems. J. Dynam. Control Systems 5 (1999), no. 4, 501--507.
A singularly perturbed system with a small parameter $\varepsilon$ at the velocity of the slow variable $y$ and with the fast variable $x$ is considered. The main hypothesis is that for all $y$ from some bounded domain $D$, the fast subsystem has a stable invariant or overflowing manifold $M_0(y)$ and that the motions in this system going in the directions transversal to $M_0(y)$ are more fast than the mutual approaching of trajectories on $M_0(y)$ (a precise statement is given in terms of appropriate Lyapunov-type characteristic numbers). It is proved that for a sufficiently small $\varepsilon$, the whole system has an invariant manifold close to \smash[b]{\bigcup\limits_{y \in D} M_0(y) \times \{y\}}; the degree of its smoothness is specifed.
M. Stephane. On Reducible Monodromies Realized by Reducible Fuchsian Systems. J. Dynam. Control Systems 5 (1999), no. 4, 509--522.
It is shown that under the conditions of Theorems~1 and 2 in [2], the Fuchsian systems realizing the reducible monodromy are in fact reducible systems. On the other hand, when the reducible monodromy is realized by a Fuchsian system, sufficient conditions on the monodromy matrices under which the final Fuchsian system can be chosen reducible are given.
V.A. Kondrat'ev. On Propertiies of Solutions to Nonlinear Parabolic Equations of the Second Order. J. Dynam. Control Systems 5 (1999), no. 4, 523--546.
A quasilinear elliptic problem is considered for which conditions for existence and nonexistence of positive solutions are discussed.
V.V. Dolotin. Groups of Flagged Homotopies and Higher Gauge Theory. J. Dynam. Control Systems 5 (1999), no. 4, 547--563.
Groups \Pi_k(X;\sigma) of ``flagged homotopies'' are introduced of which the usual (abelian for k>1) homotopy groups \pi_k(X;p) is the limit case for flags \sigma contracted to a point p. The calculus of exterior forms with values in an algebra A is developped of which the limit cases are the differential forms calculus (for A= R) and gauge theory (for 1-forms). Moduli space of integrable forms with respect to higher gauge transforms (cohomology with coefficients in A) is introduced with elements giving representations of \Pi_k in G=exp A.
V.A. Golubeva and V.P. Leksin. On Two Types of Representations of the Braid Group Associated with the Knizhnik--Zamolodchikov Equation of the B_n Type. J. Dynam. Control Systems 5 (1999), no. 4, 565--596.
A generalization of the Kohno theorem on the restricted Riemann--Hilbert problem for the KZ equation of the B_n type is given. The representation of the the braid group of the B_n type in the algebra of symmetrical chord diagrams is constructed and its connection with the B_n type quasi-bialgebra structure and with the monodromy of the generalized KZ equation of the B_n type is discussed.
|
|
|
|
of manuscripts |
|
|