Vol 52, No 11 (2016)

We realize a version of the Perron sign reversal effect for the characteristic exponents of a two-dimensional differential system; the exponents are negative for the linear approximation system and positive for the nontrivial solutions of the full nonlinear system with a higher-order perturbation in a neighborhood of the origin and with initial data on an arbitrary finite set of points and lines on the plane R².

An optimal control problem with state constraints is considered. Some properties of extremals to the Pontryagin maximum principle are studied. It is shown that, from the conditions of the maximum principle, it follows that the extended Hamiltonian is a Lipschitz function along the extremal and its total time derivative coincides with its partial derivative with respect to time.

We consider the problem of constructing manifolds of a special form, which are said to be almost integral, lie in the state space of an affine control system, and have the following property: the restriction of the control system to such a manifold is an almost trivial control system defining the part of trajectories of the original system lying on that manifold.

For linear autonomous differential-difference systems of neutral type with commensurable delays, we suggest solvability criteria for the modal controllability problem with the use of two classes of state feedback controllers, namely, controllers of constant and variable structure. The proofs are constructive and permit one to obtain the corresponding controllers with the use of standard operations on polynomials and polynomial matrices.

We consider a linear time-invariant multivariable square control system. For this system, we are interested in the description of zero dynamics, that is, the dynamics of the system for the case of identically zero output. We study the case in which the relative degree is undefined for the system.

The class of Hilbert space multicriteria optimization problems considered in the paper includes control problems for various dynamical systems with lumped as well as distributed parameters. An equilibrium point is sought under the assumption that the criteria and their derivatives are known approximately. We use a regularized extragradient method and prove its convergence. As a sample application of the general theory, we consider a control problem for a parabolic equation with two criteria.

We study the minimal and maximal operators generated by the Bessel differential expression on a finite interval and the half-line. We describe the domains of the Friedrichs and Krein extensions of the minimal operator and all nonnegative self-adjoint extensions of the minimal operator.