Vol 54, No 12 (2018)

We consider families of n-dimensional (n ≥ 2) linear differential systems on the time semiaxis with parameter varying in a metric space. For such families continuously depending on the parameter in the sense of uniform convergence on the time semiaxis, we completely describe the spectra of their Lyapunov exponents as vector functions of the parameter.

Abstract—General sufficient conditions for the existence of a solution of a boundary value problem for the φ-Laplacian with functional boundary conditions are obtained. As a consequence of this result, natural restrictions on the lower and upper functions are obtained under which there exists a solution of a periodic boundary value problem for the φ-Laplacian. Several applications of this results to the proof of the existence of solutions of boundary value problems for higher-order equations are given.

We study a system of two singularly perturbed first-order equations on an interval. The equations have discontinuous right-hand sides and equal powers of the small parameter multiplying the derivatives. We consider a new class of problems with discontinuous right-hand side, prove the existence of a solution with an internal transition layer, and construct its asymptotic approximation of arbitrary order. The asymptotic approximations are constructed by the Vasil’eva method, and the existence theorems are proved by the matching method.

We consider the polynomials T_r,n(x) (n = 0, 1,…) generated by Chebyshev polynomials T_n(x) and forming a Sobolev orthonormal system with respect to the inner product

\(\langle f,g\rangle = \sum\limits_{\nu = 0}^{r - 1} {{f^{(\nu)}}} ( - 1){g^{(\nu)}}(-1) + \int\limits_{-1}^1 {{f^{(r)}}} (x){g^{(r)}}(x)\mu (x)dx,\)

An operator model of integro-differential equations arising in the theory of viscoelasticity is studied. The spectral analysis of operator functions which are the symbols of Gurtin–Pipkin-type integro-differential equations is carried out with the Kelvin–Voigt friction taken into account.

We obtain a criterion for the unique solvability of a nonlocal boundary value problem for a second-order nonlinear Volterra integro-differential equation with degenerate kernel. Several informative examples are given.

We consider the problem of determining the equations of zero dynamics of a nonlinear system that is affine with respect to the control. All known methods for solving this problem have a restricted scope. We obtain a new algorithm for solving this problem which permits determining equations of zero dynamics of some systems to which the previously known methods cannot be applied.

We develop the internal approximation method for constructing algorithms that allow one to seek stability sets for finite families of homogeneous affine polynomials. Under certain assumptions, the complex problem of synthesis of a simultaneously stabilizing controller for a given finite family of linear stationary dynamic plants can be reduced to such a problem.