Vol 231, No 6 (2018)

For a singular Cauchy problem

\( \sum \limits_{i=0}^N\sum \limits_{j=0}^N\sum \limits_{k=0}^N{a}_{ijk}{t}^i{\left(x(t)\right)}^j{\left({x}^{\prime }(t)\right)}^k+\varphi \left(t,x(t),{x}^{\prime }(t)\right)=0\kern0.5em x(0)=0, \)

where N ≥ 2 and a_ijk are constants, a_00k = 0, k ∈ {0, 1, . . .,N} , a₁₀₀ ≠ 0, a₀₁₀ ≠ 0, a_ijk = 0, 1 ≤ i + j < m, k ∈ {1, . . . ,N} , 2 ≤ m ≤ N, and φ is a function small in a certain sense, we find a nonempty set of continuously differentiable solutions x: (0, ρ] → ℝ, where ρ is sufficiently small, such that

\( {\displaystyle \begin{array}{cc}x(t)=\sum \limits_{k=1}^m{c}_k{t}^k+o\left({t}^m\right),& t\to +0,\end{array}} \)

where c₁, . . . , c_m are known constants.

Necessary and sufficient conditions for the static output feedback stabilization of linear discrete-time control systems are formulated in the form of a matrix inequality. It is shown that the algorithms of stabilization based on these conditions are applicable to a certain class of nonlinear discrete-time control systems. We propose the procedure of construction of the regularities of control guaranteeing the required estimation of the weighted level of attenuation of input signals and initial perturbations, as well as the robust stability of the equilibrium state of the controlled system. A numerical example of the discretetime system of stabilization of a pendulum on a moving platform is presented.

We investigate the procedures of discretization of the integrable nonlinear Schrödinger dynamical system, well known as the Ablowitz–Ladik equation, the corresponding symplectic structures, and the finite-dimensional invariant reductions. We develop an efficient scheme of invariant reduction of the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter. We construct a finite set of recurrence algebraic regular relations that allows one to generate solutions of the discrete nonlinear Schrödinger dynamical system and discuss the related functional spaces of solutions. Finally, we analyze the Fourier-transform approach to the study of the set of solutions of the discrete nonlinear Schrödinger dynamical system and its functional-analytic aspects.

We study linear systems of ordinary differential equations of higher order with an identically singular matrix at the higher derivative of the required vector-valued function in the domain of definition. We give definitions of the index and singular points for these systems, formulate the conditions of solvability, and deduce a formula for the general solution. The algorithms for finding the index and singular points are presented.