Articles

For a fixed period of time, a mathematical model for the treatment of psoriasis is considered, consisting of three differential equations. These equations describe the relationships between the major cell populations that are responsible for the occurrence, progression, and treatment of this disease. The model also contains a bounded control function reflecting the effects of a drug aimed at inhibiting interactions between specific cell populations. The task is to reduce the population of cells directly affecting the disease at the end of a given period of time. The analysis of such an optimal control problem is carried out using the Pontryagin maximum principle. The main attention is paid to clarifying the bang-bang property of the corresponding optimal control, as well as the possibility of its having singular regimens of various orders depending on the relationships between the parameters of the original model.

We study conditions under which components of the distribution of the difference of two independent and identically distributed random variables are determined uniquely up to a shift and reflection. This uniqueness is essential to some characterization problems. An algorithm is presented for estimation of the components when data are given in a symmetrized form.

The paper is devoted to the development of a new method for the approximate construction of the reachability set for a nonlinear control system with discrete time. Pointwise restrictions are imposed on the control parameters. To solve this problem, a technique previously developed and applied for the case of continuous time and differential equations is used. The estimate of the reachability set can be obtained as the level set of a special piecewise affine value function constructed on a grid of simplices in the state space. The paper presents formulas for calculating the coefficients of such a function, which make it possible to analyze the difference between the case with discrete time and the case with continuous time. An example of calculation of piecewise affine value functions and corresponding internal and external estimates of the reachability set is considered.

Let A and B be matrices of order n that are direct sums of nilpotent Jordan blocks. Suppose that A and B are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, A and B cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.