Vol 53, No 4 (2017)

For a family of n-dimensional (n ≥ 2) linear differential systems depending continuously, in the sense of the uniform norm on the half-line, on a parameter varying in a complete metric space, we obtain a complete description of the sets of points of lower semicontinuity and upper semicontinuity of the ith Lyapunov exponent, i = 1,..., n, treated as a function of the parameter.

We establish some properties of the spectrum of the shift operator with nonnegative coefficients on the space of bounded functions on a locally compact commutative group and use them to study the asymptotic properties of linear discrete systems with this operator.

We consider mappings of the m-dimensional torus T^m (m ≥ 2) that are C¹-perturbations of linear hyperbolic automorphisms. We obtain sufficient conditions for such mappings to be one-to-one hyperbolic mappings (i.e., Anosov diffeomorphisms). These results are used to study the blue-sky catastrophe related to the vanishing of a saddle-node invariant torus with a quasiperiodic winding in a system of ordinary differential equations.

We consider a boundary control problem for the stationary convection–diffusion–reaction equation in which the reaction constant depends on the concentration of matter in such a way that the equation has a fifth-order nonlinearity. We prove the solvability of the boundary value problem and an extremal problem, derive an optimality system, and analyze it to derive estimates for the local stability of the solution of the extremal problem under small perturbations of both the performance functional and one of the given functions.

We consider four mixed problems for the string vibration equation with zero initial conditions, with a Bitsadze–Samarskii boundary condition of the general form at the right endpoint, and with an inhomogeneous Neumann or Dirichlet condition at the left endpoint. We prove the uniqueness of a generalized solution (in the sense of Il’in) of these problems and obtain an analytic representation of these solutions. The solution of each of the problems is represented in the form of a linear combination of functions constructed from the problem data, and recursion formulas for the coefficients of this linear combination are obtained.

We consider the one-dimensional Boltzmann equation f_t + cf_x + Ff_c = 0, where the functions f and F are assumed to depend on three variables t, x, and c. We obtain relations defining the symmetry algebra in the general case and also under the additional conditions of conservation of the relations dx = c dt and dc = F dt, which arise from physical considerations. We show that the widest symmetry algebra is obtained in the case of conservation of both relations. This algebra is infinite-dimensional, and its structure is independent of the form of the function F.

For linear autonomous completely regular differential-algebraic delay systems, we develop methods for synthesizing state feedback controllers that simultaneously provide solution damping and finite spectrum assignment in the closed-loop system. We prove necessary and sufficient conditions for the existence of such controllers defining finite- and infinite-time controls depending on the system parameters (complete 0-controllability and 0-controllability criteria, respectively). The proofs are of constructive character and allow one to synthesize the corresponding controller for every system. An illustrative example is given.

We consider a classical spectral problem that arises when studying the natural vibrations of a loaded rectangular membrane fixed on two sides, the load being distributed along one of the free sides. We study the completeness, minimality, and basis property of the system of eigenfunctions and establish conditions guaranteeing the equiconvergence of spectral expansions in this system and in a given basis.