Vol 55, No 12 (2019)

We consider families of n-dimensional (n ≥ 2) linear differential systems on the time half-line with parameter belonging to a metric space. We obtain a complete description of the Lyapunov irregularity coefficient as a function of the parameter for families whose dependence on the parameter is continuous in the sense of uniform convergence on the time half-line. As a corollary, we completely describe the parametric dependence of the Lyapunov irregularity coefficient of a regular linear system with a linear parametric perturbation decaying at infinity uniformly with respect to the parameter.

We consider a nonlinear eigenvalue problem for a system of ordinary differential equations arising in the waveguide theory. The nonlinearity is characterized by two nonnegative parameters α and β. For α = β = 0, we arrive at a linear problem that has finitely many (positive) eigenvalues. It is proved that for α > 0 and β ≥ 0 there exist infinitely many positive eigenvalues; their asymptotics is indicated. It is also proved that for α = 0 and β > 0 there exist finitely many eigenvalues. A comparison theorem for the eigenvalues is obtained for α, βs > 0. It is shown that perturbation theory methods cannot be used to study the nonlinear problem completely.

We study the behavior of solutions of Volterra type systems for various versions of asymptotic localization of the Malthusian vector. In particular, it is proved that if the origin belongs to the boundary of a. convex set that is almost attracting for such a. vector calculated on a. solution and if the solutions are logarithmically bounded, then the minimum face of this set containing the origin is almost attracting for this vector as well. Applications of the results to problems arising in mathematical biology are considered.

We consider the self-adjoint operator L (the Friedrichs extension) associated with the closable form \({a_m}[u,f] = \int_\Omega {\left( {\sum\nolimits_{\left| \alpha \right| = m} {{\rho ^2}(x){D^\alpha }u\overline {{D^\alpha }f} + v_m^2(x)u\bar f} } \right)dx}\) in the space L_2,ω where Ω ⊂ ℝⁿ is a domain, \(f,u \in C_0^\infty (\Omega )\), and m ∈ ℕ. Here v_s(x) = ρ(x)h^-s(x), s > 0, and the positive functions ρ(·) and h(·) satisfy some special conditions. The space L_2,ω is a weighted Hilbert space with a locally integrable weight ω(x) positive almost everywhere in Ω. For s > 0 and 1 < p < ∞, we introduce the weighted space of potentials \(H_p^s(\rho ,{v_s})\). For s = m ∈ ℕ and p = 2, the space \(H_2^m(\rho ,{v_m})\) is the weighted Sobolev space \(H_2^m(\rho ,{v_m})\) with the equivalent norm \(\left\| {f;W_2^m(\rho ,{v_m})} \right\| = \sqrt {{a_m}[f,f]}\). Descriptions are given of the interpolation spaces H_s obtained by the complex and real interpolation methods from the pair \(\left\{ {H_p^{{s_0}}(\rho ,{v_{{s_0}}}), H_p^{{s_1}}(\rho ,{v_{{s_1}}})} \right\}\), 0 < s₀ < s₁. Estimates are derived for the linear widths and Kolmogorov widths of the compact sets \(\mathcal{F} = B{H_s}\bigcap {{L^{ - 1}}(B{L_{2,\omega }})}\), s > 0 (where BX is the unit ball in a space X).

For a spectrally controllable and simultaneously spectrally observable linear time-invariant system with commensurable delays, we construct one-dimensional output feedback controls that ensure the complete stabilization of the closed-loop system (the complete damping of the original system and the asymptotic stability of the closed-loop system) by an integro-difference controller and the finite-time stabilization (the complete damping of the original system) by a differential-difference controller. The results are illustrated with an example.

We study the problem of stabilizing multiple-input switched linear systems that may have operation modes of different dynamic orders. To solve this problem, we propose two approaches to constructing a. stabilizing controller based on the dynamic-order extension method and on solving a. system of linear matrix inequalities.

We consider the center-focus problem for a cubic system that generalizes the system considered in Differentsial’nye Uravneniya, 1978, vol. 14, no. 2, pp. 380–382.