### Book chapter

## Optimal Solutions in a Neighborhood of a Singular Extremal for a Problem with Two-Dimensional Control

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.

### In book

Lax pairs for linear Hamiltonian systems of differential equations are found in the paper. The first integrals of the system obtained from these Lax pairs are investigated.

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.

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.

We consider a linear model of a rotating Timoshenko beam. We show that for some initial conditions, the solutions of the minimization problem for the deviation of the beam from the equilibrium state have Fuller singularities.

A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold M and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field Xwith a global cross-section there exist some 4-dimensional symplectic manifold M̃⊃M and a smooth Hamilton function H:M̃→R such that for some c∈R one gets M={H=c}and the Hamiltonian vector field XH restricted on this level coincides with X. For divergence free vector fields with singular points such an extension is impossible but the existence of local cross-section allows one to reduce the dynamics to the study of symplectic diffeomorphisms in some sub-domains of M. We also consider the case of a divergence free vector field X with a smooth integral having only finite number of critical levels. It is shown that such a noncritical level is always a 2-torus and restriction of X on it possesses a smooth invariant 2-form. The linearization of the flow on such a torus (i.e. the reduction to the constant vector field) is not always possible in contrast to the case of an integrable Hamiltonian system but in the analytic case (M and X are real analytic), due to the Kolmogorov theorem, such a linearization is possible for tori with Diophantine rotation numbers.

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.