Vol 29, No 145 (2024)

The well-established criterion for the action and boundedness of a linear integral operator K from the space $L_{\infty }$ of essentially bounded functions to the space C of functions continuous on a compact set is extended to the case of functions taking values in Banach spaces.

The study further shows that if the operator K is active and bounded in the space C, it is also active and bounded in the space $L_{\infty }\mathrm{,}$ with the norms of K in C and $L_{\infty }$ being identical. A precise expression for the general value of the norm of the operator K in these spaces, expressed in terms of its operator kernel, is provided. Addicionally, an example of an integral operator (for scalar functions) is given, active and bounded in each of the spaces C and $L_{\infty },$ but not acting from $L_{\infty }$ into C.

Convenient conditions for checking the boundedness of the operator K in C and $L_{\infty }$ are discussed. In the case of the Banach space Y of the image function values of K being finite-dimensional, these conditions are both necessary and sufficient. In the case of infinite-dimensionality of Y they are sufficient but not necessary (as proven).

For $\dim Y<\infty ,$ unimprovable estimates for the norm of the operator K are provided in terms of a 1-absolutely summing constant ${\pi }_1\left(Y\right),$ determined by the geometric properties of the norm in Y. Specifically, it is defined as the supremum over finite sets of nonzero elements of Y of the ratio of the sum of the norms of these elements to the supremum (over functionals with unit norm) of the sums of absolute values of the functional on these elements.

The work is devoted to the issues of mathematical modeling of the spatial motion of a quadcopter and the construction of program control laws that ensure flight with the values of part of the coordinates of the phase vector specified at intermediate times. A structural diagram of a quadcopter with four propeller engines is used, which allows for movement in space, vertical takeoff and landing. Based on the laws of theoretical mechanics, a system of differential equations is obtained that describes the spatial motion of such a quadcopter. For a linearized mathematical model of quadcopter motion, the problem of constructing program control laws with given initial and final values of the phase vector, as well as the values of part of the coordinates of the phase vector at two intermediate moments of time, has been solved. A necessary and sufficient condition for the existence of program control is obtained and the corresponding movement of the quadcopter is described. Control functions and corresponding phase trajectories of motion are constructed. To illustrate the results obtained, for specific initial, final and intermediate values, explicit expressions of the program control function, program motion are obtained and the corresponding graphs are constructed.

For an equality-constrained optimization problem, we consider the possibility to interpret sequential quadratic programming methods employing the Hessian of the Lagrangian reduced to the null space of the constraints’ Jacobian, as a perturbed Newton–Lagrange method. We demonstrate that such interpretation with required estimates on perturbations is possible for certain sequences generated by variants of these methods making use of second-order corrections. This allows to establish, from a general perspective, superlinear convergence of such sequences, the property generally missing for the main sequences of the methods in question.

We consider an autonomous system of differential equations $\dot{x}=f\left(x\right),wherex\in {ℝ}^n,$ the vector function $f\left(x\right)$ and its derivatives $\partial f_i\partial x_j$ ($i,j=\mathrm{1,}\dots ,n$) are continuous. Three classes of autonomous systems are identified and the properties that systems of each class possess are described.

We will assume that the system belongs to the first class on the set $D\subseteq {ℝ}^n\mathrm{,}$ if the right parts of this system do not depend on varibles $x_1\mathrm{,}\dots \mathrm{,}{x}_n\mathrm{,}$ that is this system has the form $\dot{x}=C,$ where $C\in {ℝ}^n\mathrm{,}$ $x\in D\mathrm{.}$ We will assign to the second class the systems that are not included in the first class, for which the next condition is met “each of the function $f_i$ is increasing on the set $D\subseteq {ℝ}^n$ with respect to all variables on which it explicitly depends, with the exception of variable $x_i\mathrm{,}$ $i=\mathrm{1,}\dots ,n$”. Solutions of systems of the first and second classes have the property of monotonicity with respect to initial conditions.

We will assign to the third class the systems that are not included in the first class, for which the condition is met “each of the function $f_i$ is decreasing on the set $D\subseteq {ℝ}^n$ with respect to all variables on which it explicitly depends, with the exception of variable $x_i\mathrm{,}$ $i=\mathrm{1,}\dots ,n$”.

The conditions for the absence of periodic solutions for autonomous systems of the second order are obtained, complementing the known Bendikson conditions. It is proved that systems of two differential equations of all three specified classes cannot have periodic solutions

The task of obtaining the sharp estimate of the modulus of the n-th Taylor coefficient on the class B of bounded non-vanishing functions has been reduced to the problem of estimating a functional over the class of normalized bounded functions, which in turn has been reduced to the problem of finding the constrained maximum of a non-negative objective function of $2n-3$ real arguments with constraints of the inequality type, that allows us to apply the standard numerical methods of finding constrained extrema.

Analytical expressions of the first six objective functions have been obtained and their Lipschitz continuity has been proved. Based on the Lipschitz continuity of the objective function with number n, a method for the sharp estimating of the modulus of the n-th Taylor coefficient on the class B is rigorously proven. An algorithm of finding the global constrained maximum of the objective function is being discussed. The first step of this algorithm involves a brute-force search with a relatively large step. The second step of the algorithm uses a method for finding a local maximum with the initial points obtained at the previous step.

The results of the numerical calculations are presented graphically and confirm the Krzyz conjecture for $n=\mathrm{1,}\dots \mathrm{,6.}$ Based on these calculations, as well as on so-called asymptotic estimates, a sharp estimate of the moduli of the first six Taylor coefficients on the class B is derived. The obtained results are compared with previously known estimates of the moduli of initial Taylor coefficients on the class B and its subclasses $B_t\mathrm{,}$ $t⩾0.$ The extremals for  subclasses are discussed and the Krzyz hypothesis is updated for $B_t$ subclasses. A brief historical overview of research of the estimations of moduli of initial Taylor coefficients on the class B is provided.