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## Оптимальный синтез в бесконечномерном пространстве

For a class of optimal control problems and Hamiltonian systems generated by these problems in the space *l *2, we prove the existence of extremals with a countable number of switchings on a finite time interval. The optimal synthesis that we construct in the space *l *2 forms a fiber bundle with piecewise smooth two-dimensional fibers consisting of extremals with a countable number of switchings over an infinite-dimensional basis of singular extremals.

Energy-saving optimization is very important for various engineering problems related to modern distributed systems. We consider here a control problem for a wireless sensor network with a single time server node and a large number of client nodes. The problem is to minimize a functional which accumulates clock synchronization errors in the clients nodes and the energy consumption of the server over some time interval [0,T]. The control function u=u(t), 0\leq u(t)\leq u_{1}, corresponds to the power of the server node transmitting synchronization signals to the clients. For all possible parameter values we find the structure of extremal trajectories. We show that for sufficiently large u_{1} the extremals contain singular arcs.

We study singularities of optimal solutions in a problem of controlling the Timoshenko beam vibrations. The Timoshenko beam vibrations are described by a system of two coupled hyperbolic equations. Controls are introduced as external bounded forces. We consider the problem of minimizing the mean square deviation of the Timoshenko beam from the equilibrium position. For some initial conditions we reduce this problem to the optimal control problem for ordinary differential equations. We study the case of two-dimensional controls. For some initial positions of the beam, we prove that the optimal solutions have a Fuller type singularity. We give an asymptotic representation of the corresponding family of optimal trajectories.

We consider an optimal control problem that is affine in two-dimensional bounded control. We study a behavior of solutions in a neighborhood of a singular extremal. We show that there exists optimal spiral-similar solution which attains the singular point in finite time making a countable number of rotations.

Power consumption, clock synchronization and optimization are very popular topics an analysis of wireless sensor networks. In the present talk we consider a mathematical model related with large scale networks which nodes are equipped with noisy non-perfect clocks. The task of optimal clock synchronization in such networks is reduced to the classical control problem. Its functional is based on the trade-off between energy consumption and mean-square synchronization error. This control problem demonstrates surprisingly deep connections with the theory of singular optimal solutions.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.