Vol 58, No 12 (2018)

A terminal control problem with linear controlled dynamics on a fixed time interval is considered. A boundary value problem in the form of a linear programming problem is stated in a finite-dimensional terminal space at the right endpoint of the interval. The solution of this problem implicitly determines a terminal condition for the controlled dynamics. A saddle-point approach to solving the problem is proposed, which is reduced to the computation a saddle point of the Lagrangian. The approach is based on saddle-point inequalities in terms of primal and dual variables. These inequalities are sufficient optimality conditions. A method for computing a saddle point of the Lagrangian is described. Its monotone convergence with respect to some of the variables on their direct product is proved. Additionally, weak convergence with respect to controls and strong convergence with respect to phase and adjoint trajectories and with respect to terminal variables of the boundary value problem are proved. The saddle-point approach is used to synthesize a feedback control in the case of control constraints in the form of a convex closed set. This result is new, since, in the classical case of the theory of linear regulators, a similar assertion is proved without constraints imposed on the controls. The theory of linear regulators relies on matrix Riccati equations, while the result obtained is based on the concept of a support function (mapping) for the control set.

The existence, stability, and asymptotic behavior of steady modes for a delay logistic equation with slowly varying coefficients are analyzed.

The propagation of a diffusion–reaction plane traveling wave (for example, a flame front), the charge distribution inside a heavy atom in the Thomas–Fermi model, and some other models in natural sciences lead to bounded solutions of a certain autonomous nonlinear second-order ordinary differential equation reducible to an Abel equation of the second kind. In this study, a sufficient condition is obtained under which all solutions to a special second-kind Abel equation that pass through a nodal singular point of the equation can be represented by a convergent power series (in terms of fractional powers of the variable) in a neighborhood of this point. Under this condition, new parametric representations of bounded solutions to the corresponding autonomous nonlinear equation are derived. These representations are efficient for numerical implementation.

A family of exact solutions of an evolution equation describing the combustion process in a medium with a power-law temperature dependence of the source density is found. A formal asymptotics of the solution of the initial boundary value problem for the reaction–diffusion equation is constructed. The correctness of the partial sum of an asymptotic series is proved using the method of differential inequalities.

Existence and uniqueness theorems for inverse problems of determining the right-hand side and lowest coefficient in a degenerate parabolic equation with two independent variables are proved. It is assumed that the leading coefficient of the equation degenerates at the side boundary of the domain and the order of degeneracy with respect to the variable \(x\) is not lower than 2. Thus, the Black–Scholes equation, well-known in financial mathematics, is admitted. These results are based on the study of the unique solvability of the corresponding direct problem, which is also of independent interest.

The inverse problem of determining a temperature-dependent thermal conductivity coefficient is studied. The study is based on the Dirichlet boundary value problem for the two-dimensional nonstationary heat equation. The cost functional is defined as the rms deviation of the surface heat flux from experimental data. For the numerical solution of the problem, an algorithm based on the modern fast automatic differentiation technique is proposed. Examples of solving the posed problem are given.

The solvability of boundary-value and extremum problems for a nonlinear convection–diffusion–reaction equation with mixed boundary conditions is proved in the case where the coefficient in the boundary condition is a fairly arbitrary function of the solution to the boundary value problem. For the mass transfer coefficient equal to the modulus of the substance concentration, local stability estimates of the solution to the extremum problem with respect to relatively small perturbations in the cost functional and the given functions of the boundary value problem are obtained.

We consider a strongly NP-hard problem of partitioning a finite Euclidean sequence into two clusters of given cardinalities minimizing the sum over both clusters of intracluster sums of squared distances from clusters elements to their centers. The center of one cluster is unknown and is defined as the mean value of all points in the cluster. The center of the other cluster is the origin. Additionally, the difference between the indices of two consequent points from the first cluster is bounded from below and above by some constants. A randomized algorithm that finds an approximation solution of the problem in polynomial time for given values of the relative error and failure probability and for an established parameter value is proposed. The conditions are established under which the algorithm is polynomial and asymptotically exact.