Vol 53, No 8 (2017)

We present an explicit form of cubic systems with a nilpotent singular point of the focus or center type at the origin. A method for finding the focus quantities of such systems is indicated. Sufficient conditions for the existence of a nilpotent center for cubic systems are given. Cubic systems reducible to the Li´enard system are studied in detail.

We use the variational method to establish criteria for the existence of conjugate points and for the oscillation property of the linear differential Sturm–Liouville equation.

We consider a relay system of ordinary differential equations whose right-hand sides are sums of linear functions and two discontinuous functions. We analyze the stability of the zero solution of a relay system of this form for the case in which the system parameters satisfy some equality-type constraints.

We consider the Sturm–Liouville operator generated in the space L²[0,+∞) by the expression l_a,b:= −d²/dx² +x+aδ(x−b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ_n of this operator satisfy the inequalities λ₁⁰ < λ₁ < λ₂⁰ and λ_n⁰ ≤ λ_n < λ_n+1⁰, n = 2, 3,..., where {−λ_n⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.

We study a boundary value problem for a nonlinear equation of mixed type with the Lavrent’ev–Bitsadze operator in the principal part and with functional delay and advance in lower-order terms. The general solution of the equation is constructed. The problem is uniquely solvable.

We describe the phase space of the first initial–boundary value problem for a system of partial differential equations modeling the motion of an incompressible viscoelastic Kelvin–Voigt fluid of nonzero order in the Earth magnetic field. In the framework of the theory of semilinear equations of Sobolev type, we prove the existence and uniqueness of a solution that is a quasistationary semitrajectory.

We study the solvability of the inverse nonlinear system analysis problem viewed as qualitative solvability (necessary and sufficient conditions) of a realization of a time-varying polylinear controller as a second-order differential system whose admissible solutions include a given nonlinear pencil of arbitrary (finite, countable, or continual) cardinality of dynamic processes in a separable Hilbert space.

We consider the problem of designing a digital controller stabilizing a continuoustime switched linear system. Our approach to stabilization includes the construction of a continuous-discrete time closed-loop system, the passage to its discrete-time model, and the subsequent discrete-time controller design based on simultaneous stabilization methods.

We consider the Dirichlet problem with nonclassical boundary conditions for a 2nth-order pseudoparabolic equation with nonsmooth coefficients in a rectangular parallelepiped. We prove that this problem is equivalent to the classical Dirichlet problem.