Vol 59, No 12 (2023)

For a nonlinear differential matrix equation, we study a multipoint boundary value
problem by a constructive method of regularization over the linear part of the equation using
the corresponding fundamental matrices. Based on the initial data of the problem, sufficient
conditions for its unique solvability are obtained. Iterative algorithms containing relatively
simple computational procedures are proposed for constructing a solution. Effective estimates
are given that characterize the rate of convergence of the iteration sequence to the solution, as
well as estimates of the solution localization domain.

We consider the solution of the Cauchy problem in a strip on the plane for a homogeneous
second-order parabolic system. The coefficients of the system satisfy the double
Dini condition. The initial function is continuous and bounded along with its first and second
derivatives. Using the Poisson potential, the nature of the smoothness of this solution is studied
and the corresponding estimates are proved.

We consider initial–boundary value problems for homogeneous parabolic systems with coefficients satisfying the double Dini condition with zero initial conditions in a semibounded
plane domain with nonsmooth lateral boundary. The method of boundary integral equations is used to prove a theorem on the unique classical solvability of such problems in the space of functions that are continuous together with their first spatial derivative in the closure of the domain. An integral representation of the obtained solutions is given. It is shown that the condition for the solvability of the posed problems considered in the paper is equivalent to the well-known complementarity condition.

The inverse spectral problem method is applied to finding a solution of the Cauchy
problem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.
A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded terms
and a derivation of an analog of Dubrovin’s system of differential equations are proposed. It
is shown that the sum of a uniformly convergent function series constructed by solving the
Dubrovin system of equations and the first trace formula actually satisfies the loaded nonlinear
Korteweg–de Vries equation. In addition, we prove that if the initial function is a real π-periodic
analytic function, then the solution of the Cauchy problem is a real analytic function in the
variable x as well, and also that if the number π/n, n ∈ N, n ≥ 2, is the period of the initial
function, then the number π/n is the period for solving the Cauchy problem with respect to the
variable x.

We study the problem of optimal choice of model parameters with respect to any
functional. Locally controllable affine systems and integral functionals depending on the program
control are considered. Local controllability of affine systems with nonnegative inputs is
proved in the case where the columns multiplying the controls form a positive basis. For such
systems, we introduce the local controllability coefficient and pose the problem of its maximization
depending on the choice of model parameters. As an example, we consider a very simplified
model of an underwater vehicle and study the problem of finding an arrangement of its control
propellers in which the energy consumption of the vehicle is minimal.

We study the feedback control problem for a mathematical model that describes
the motion of a viscoelastic fluid with memory along the trajectories of the velocity field. We
prove the existence of an optimal control that delivers a minimum to a given bounded and lower
semicontinuous cost functional.