Vol 52, No 8 (2016)

For nontrivial solutions of a linear nonautonomous differential equation with integrally small coefficients, we improve earlier-known upper bounds for the wandering rate. In particular, our estimates imply that the upper bound of the range of the wandering rate for equations of arbitrary order tends to zero as all of their coefficients uniformly (on the time half-line) tend to zero at infinity.

We prove a comparison theorem for the solutions of Riccati matrix equations in which the diagonal entries of the matrix multiplying the linear term are perturbed by a bounded function. This theorem is used to study optimal trajectories in a pollution control problem stated in the form of a linear regulator over an infinite time horizon with a discount function of the general form.

We consider the Dirac operator on a finite interval with a potential belonging to some set X completely bounded in the space L₁[0, π] and with strongly regular boundary conditions. We derive asymptotic formulas for the eigenvalues and eigenfunctions of the operator; moreover, the constants occurring in the estimates for the remainders depend on the boundary conditions and the set X alone.

We prove the existence and uniqueness of an energy class solution of an initial–boundary value problem for a semilinear equation in divergence form. We consider the case in which an inhomogeneous third boundary condition is posed on one part of the lateral surface of the cylinder in which the equation is studied and the homogeneous Dirichlet boundary condition is posed on the other part of the lateral surface.

We obtain sufficient conditions for a certain estimate to hold on the half-line for all solutions of a linear homogeneous Volterra integro-differential equation of the first order in the critical case. We present some corollaries concerning the absolute integrability of powers of the solutions on the half-line and the convergence of solutions to zero (including the exponential and power-law convergence) as the independent variable tends to infinity. Illustrative examples are given.

We study the input tracking problem for a parabolic equation on an infinite time interval on the basis of the measurement of phase coordinates. We suggest an algorithm stable under information noises and roundoff errors for the solution of the problem on the basis of constructions of dynamic inversion theory.

We suggest a generalization of the classical notion of relative degree for linear dynamical MIMO systems. We study the properties of that notion, its relationship with the zero dynamics of the system, and the possibility of reduction of the system to a special form.

We describe all degenerate boundary conditions in the homogeneous Sturm–Liouville problem.

We find closed-form recursion formulas for the unique classical solution of a mixed problem describing forced vibrations of a bounded string under two boundary modes with timedependent oblique derivatives. The formulas do not use any continuation of the problem data. We obtain conditions on the right-hand side of the equation necessary and sufficient for the well-posed global solvability of the problem.