Vol 239, No 2 (2019)

We consider a linear nonstationary system of ordinary differential equations with interval coefficients which is not solvable with respect to the derivative of the unknown vector-valued function for any matrix coefficients in a given interval family. We obtain conditions sufficient for the preservation of the internal structure of the system. Under conditions guaranteeing the structure preservation, we obtain sufficient and necessary robust stability conditions. A variable rank of matrix coefficients and an arbitrary high unsolvability index are allowed.

For a nonlinear parabolic heat equation we construct a heat wave type solution composed of the zero and nonnegative solutions joined continuously along the wave front. We prove the existence and uniqueness of an analytic solution to the problem with a given wave front in the cases of plane, circular, and spherical symmetry. The solution is constructed in the form of a characteristic series with recurrently defined coefficients. In the case of a power source, we show that the original problem can be reduced to the Cauchy problem for a second order ordinary differential equation and the solution is invariant. We present numerical results verified by using the constructed analytic solutions.

We consider a semilinear differential-algebraic system of partial differential equations of index (k, 0). We numerically solve this system by applying the spline-collocation method based on splitting the matrix pencil. The method has high accuracy coinciding with the smallest order of the approximating spline.

An optimal control problem for a system of one dimensional hyperbolic conservation laws is considered. The system is controlled through the boundary condition, which depends on the state of a nonlinear control system. It is proven that the problem has an optimal solution; moreover, it admits ε-optimal solutions generated by bang-bang controls. Bibliography: 8 titles. Illustrations: 1 figure.

We study two game families with total payoffs that are defined either as the Cesàro average (the long run average game family) or Abel average (the discounting game family) of the running costs. We study value functions for all sufficiently small discounts and for all sufficiently large planning horizons (asymptotic approach), investigate a robust strategy that provides a near-optimal total payoff in this case (uniform approach). Assuming the Dynamic Programming Principle, we prove the corresponding Tauberian theorems without requiring the boundedness of the running cost.