Vol 55, No 3 (2019)

The limit frequency properties of trajectories of the simplest dynamical system generated by the left shift on the space of sequences of letters from a finite alphabet are studied. The following modification of the Shannon-McMillan-Breiman theorem is proved: for any invariant (not necessarily ergodic) probability measure μ on the sequence space, the logarithm of the cardinality of the set of all μ-typical sequences of length n is equivalent to nh(μ), where h(μ) is the entropy of the measure μ. Here a typical finite sequence of letters is understood as a sequence such that the empirical measure generated by it is close to μ (in the weak topology).

We obtain a complete description of vector functions whose components are the Lyapunov and Perron irregularity coefficients of linear differential systems continuously depending on a parameter varying in a metric space.

We consider the analytic properties of solutions of the fourth-order higher analog of the second Painlevé equation and study the local properties of solutions, Bäcklund transformations, rational solutions, and their representation via generalized Yablonskii-Vorob’ev polynomials.

for the one-dimensional wave equation given in a half-strip, we consider a boundary value problem with the Cauchy conditions and two nonlocal integral conditions. Each integral condition is a linear combination of a linear Fredholm integral operator along the lateral side of the half-strip applied to the solution and the values of the solution and of a given function on the corresponding base of the half-strip. Under the assumption of appropriate smoothness of the right-hand side of the equation and the initial data, we obtain a necessary and sufficient condition for the existence and uniqueness of a classical solution of this problem and propose a method for finding it in analytical form. A classical solution is understood as a function that is defined everywhere in the closure of the domain where the equation is considered and has all classical derivatives occurring in the equation and in the conditions of the problem.

We consider the problems of optimal control of a dynamical system whose right-hand side is discontinuous in the state variable and is linear in the control with sufficiently smooth coefficients in each of the half-spaces into which the space is divided by the switching hyperplane. The main attention is paid to the situation where there exist intervals on which the optimal trajectory lies on the switching surface. New nondegenerate necessary conditions for optimality are stated and proved in the maximum principle form. The obtained optimality conditions are compared with the already known conditions.

Two types of observers are proposed for linear autonomous differential-difference systems of neutral type. The first observer can be applied to completely identifiable systems (generalization of the spectral observability property to systems of neutral type), and it asymptotically exactly reconstructs the current state of the system. The second observer can be applied to systems without the above-cited property. The estimation error of the second observer is described, and conditions under which it asymptotically exactly reconstructs the current state of the system are proposed, as well as conditions under which the estimation error is bounded. The results are illustrated with examples.

We consider the initial-boundary value problem for quasilinear parabolic equation with mixed derivatives and an unbounded nonlinearity. We construct unconditionally monotone and conservative finite-difference schemes of the second-order accuracy for arbitrary sign alternating coefficients of the equation. For the finite-difference solution, we obtain a two-sided estimate completely consistent with similar estimates for the solution of the differential problem, and also obtain an important a priori estimate in the uniform C-norm. These estimates are used to prove the convergence of finite-difference schemes in the grid L₂-norm. All theoretical results are obtained under the assumption that some conditions imposed only on the input data of the differential problem are satisfied.