Derev’yanko N. V.

We establish necessary conditions for the optimality of smooth boundary and initial controls in a semilinear hyperbolic system of the first order. The problem adjoint to the original problem is a semilinear hyperbolic system without initial conditions. The suggested approach is based on the use of special variations of continuously differentiable controls. The existence of global generalized solutions for a semilinear first-order hyperbolic system in a domain unbounded in time is proved. The proof is based on the use of the Banach fixed-point theorem and a space metric with weight functions.

We establish upper estimates for the approximation of the classes $H_p^{Ω}$ of periodic functions of many variables by polynomials constructed by using the system obtained as the tensor product of the systems of functions of one variable. These results are then used to establish the exact-order estimates of the orthoprojective widths for the classes $H_p^{Ω}$ in the space $L_p$ with $p ∈ \{1, ∞\}$.

We obtain exact-order estimates for the trigonometric widths of the classes $B^{\Omega}_{p\theta}$ of periodic functions of many variables in the space $L_q$ for some relations between the parameters $p$ and $q$.