Vol 28, No 143 (2023)
Original articles
Estimation of total income with discounting for probabilistic models of population dynamics
Abstract
217-226
On the existence of admissible processes for control systems with mixed constraints
Abstract
A control system with mixed equality-type constraints and end-point constraints is considered. In terms of the generalized Jacobian (Clarke’s derivative) with respect to the control variable of the mapping defining the constraints, sufficient conditions for the existence of continuous admissible positional controls are obtained. The proof of the corresponding theorem is based on reducing the control system to a boundary value problem for an ordinary differential equation via a nonlocal implicit function theorem. This problem is then reduced to the problem of finding a fixed point of a continuous mapping defined on a finite-dimensional closed ball and to applying an analogue of Brouwer’s fixed point theorem. In addition, a control system with mixed inequality-type constraints and end-point constraints is studied. In terms of the first derivatives with respect to the control variable of the functions that define the constraints, sufficient conditions for the existence of continuous admissible positional controls are also obtained. The proof of the corresponding theorem is carried out by passing from a system of smooth inequality-type constraints to one locally Lipschitz equality-type constraint.
227-235
Mathematical modeling in the problem of developing an effective method for controlling fusarium of wheat ear
Abstract
In this paper we constructed a mathematical model based on a continuous dynamic system, which is formalizing the interaction of fusarium fungi, wheat plants and soil microorganisms (mycophages and saprophages). The paper presents a statistical analysis of the available experimental data obtained under laboratory conditions, on the basis of which we solved the problem of restoring biologically interpreted parameters of the constructed model of the considered ecological system. The paper also considers the problem of impulse control within the constructed mathematical framework, which corresponds to a correction on the food webs in the system in order to stimulate the growth of the populations of natural antagonists of the fusarium fungus causing wheat pathology by applying special mixtures of organic fertilizers to the soil. We obtained conditions that guarantee controllability within the framework of the constructed mathematical model, as well as providing continuous dependence of solutions of the modeling equations on control.
236-244
Dynamic evaluation of criteria of the health-relatad quality of life on the base of the hierarchies analysis method
Abstract
A method for studying the dynamics of the main criteria of the health-related quality of life (HRQoL), based on the hierarchies analysis method (HAM) of T. Saaty is being developed. It is assumed that the assessments of HRQoL scales (criteria) according to existing specific and non-specific questionnaires change over time, which also leads to a change in the degree of their influence on each other. This, in turn, changes the integral indicators of HRQoL. There is a need to work with tables of dynamic judgments based on HAM using different classes of time functions, considering the specifics of the task and HRQoL criteria. The paper proposes a new methodology for estimating HRQoL, taking into account the dynamics of the main characteristics of HAM.
245-255
To probabilistic description of an ensemble of trajectories to a continuous-discrete control system with incomplete information
Abstract
A linear control system with continuous and discrete times and discrete memory is considered. The model includes an uncertainty in the description of operators implementing control actions. This uncertainty is a consequence of random disturbances under the assumption of their uniform distribution over known intervals. With each implementation a corresponding trajectory arises from random perturbations, and in the aggregate - an ensemble of trajectories, for which a component-by-component probabilistic description is given in the form of a set of probability density functions parametrized by the current time. To construct these functions, the previously obtained representation of the Cauchy operator of the system under consideration is used. The proposed probabilistic description of perturbations for trajectory variables allows one to find their standard characteristics, including expectation and variance, as well as the entire possible range of values. The results are constructive in nature and allow for effective computer implementation. An illustrative example is given.
256-267
Ekeland variational principle for quasimetric spaces
Abstract
In this paper, we study real-valued functions defined on quasimetric spaces. A generalization of Ekeland’s variational principle and a similar statement from the article [S. Cobzas, “Completeness in quasi-metric spaces and Ekeland Variational Principle”, Topology and its Applications, vol. 158, no. 8, pp. 1073–1084, 2011] is obtained for them. The modification of the variational principle given here is applicable, in particular, to a wide class of functions unbounded from below. The result obtained is applied to the study the minima of functions defined on quasimetric spaces. A Caristi-type condition is formulated for conjugate-complete quasimetric spaces. It is shown that the proposed Caristi-type condition is a sufficient condition for the existence of a minimum for lower semicontinuous functions acting in conjugate-complete quasimetric spaces.
268-276
The coefficient problem for bounded functions and its applications
Abstract
A review of the solution of the classical coefficient problem on the class $\Omega_0$ of bounded in the unit disc functions $\omega$ with normalization $\omega(0)=0,$ going back to I. Schur, is given. Then the first six inequalities, describing respectively the first six coefficient bodies on the class $\Omega_0,$ are derived. Next, a method of obtaining similar inequalities for classes $M_F$ of functions subordinated to the holomorphic function $F,$ giving the solution of the coefficient problem for these classes, is given. Then the properties of the mentioned inequalities as well as the relations between them are analyzed. In addition, it is shown that only one $n$-th inequality is sufficient to describe the $n$-th body of coefficients on the class $\Omega_0,$ and hence on $M_F.$
The problems of estimating both the modulus of each initial Taylor coefficient individually and estimating modules of all Taylor coefficients at once are discussed.
The problem of obtaining the sharp estimates of the modulus of the Taylor coefficient with number $n,$ i.e. the functional $|\{f\}_n|,$ on the class $M_F$ is at first reduced to the problem of estimating the functional over the class $\Omega_0,$ which in turn is reduced to the problem of finding the maximal modulo of conditional extremum of a real-valued function of $2(n-1)$ real arguments with constraints of inequality type $0 \leqslant x_k \leqslant1,$ $0\leqslant\varphi_k<2\pi,$ which allows us to apply standard methods of differential calculus to study for extrema, since the target function is infinitely smooth in all of its arguments. For this purpose, the results of the solution of the classical coefficient problem on the class $\Omega_0$ are used.
277-297
Regularization of classical optimality conditions in optimization problems for linear Volterra-type systems with functional constraints
Abstract
e consider the regularization of classical optimality conditions (COCs) — the Lagrange principle (LP) and the Pontryagin maximum principle (PMP) — in a convex optimal control problem with functional constraints such as equalities and inequalities. The controlled system is given by a linear functional-operator equation of the second kind of general form in the space , the main operator on the right side of the equation is assumed to be quasi-nilpotent. The problem functional to be minimized is convex (probably not strongly). The regularization of the COCs in the non-iterative and iterative forms is based on the use of the methods of dual regularization and iterative dual regularization, respectively. Obtaining non-iterative regularized COCs uses two regularization parameters, one of which is “responsible” for the regularization of the dual problem, the other is contained in a strongly convex regularizing Tikhonov addition to the objective functional of the original problem, thereby ensuring the correctness of the problem of minimizing the Lagrange function. The main purpose of regularized LP and PMP is the stable generation of minimizing approximate solutions (MASs) in the sense of J. Warga. Regularized COCs: 1) are formulated as existence theorems for MASs in the original problem with simultaneous constructive representation of specific MASs; 2) are sequential generalizations of classical analogues — their limiting variants and preserve the general structure of the latter; 3) “overcome” the ill-posedness properties of the COCs and give regularizing algorithms for solving optimization problems. Illustrating examples are considered: the problem of optimal control for the equation with delay, the problem of optimal control for the integrodifferential equation of the type of transport equation.
298-325
Estimates of the phase trajectories of controlled systems with multi-valued impulses
Abstract
We consider a controlled system for the differential equation where the parameter $\xi$ is an element of some given metric space, the control $u$ satisfies the constraint It is assumed that at each given moment of time $t_k\in (a,b)$ a solution $x:[a,b]\to \mathbb{R}^n$ (a phase trajectory) suffers discontinuity, the magnitude of which belongs to a non-empty compact set $I_k( x(t_k))\subset \mathbb{R}^n,$ and is an absolutely continuous function on intervals $(t_{k-1},t_k]$. The control function is assumed to be measurable. A theorem on estimating the distance from a given piece-wise absolutely continuous function $y:[a,b]\to \mathbb{R}^n$ to the set of phase trajectories for all initial values from a neighborhood of a vector $x_0$ and for all parameters from a neighborhood of a point $\xi_0$ is proven. It is assumed that for the given initial value $\mathrm{x}=x_0$ of the solution and for the value $\xi=\xi_0$ of the parameter, the set of phase trajectories is a priori limited. The proven theorem allows, by selecting the function $y$, to obtain an approximate solution of the controlled system, as well as an estimate of the error of such solution.
326-334
About topological properties of attraction set in ultrafilter space
Abstract
The representation of attraction set (AS) in the class of nets in the ultrafilter space on the broadly understood measurable space (MS) with topologies of Stone and Wallman types is considered. Representation of the interior of AS and some of its implications are obtained. Possibilities of the choice of usual solutions are defined by specifying constraints of asymptotic nature (CAN). The mentioned CAN can be connected with weakening of standard constraints (in control problems, boundary and intermediate conditions, phase restrictions; in problems of mathematical programming, constraints of inequality type), but they may appear initially in the form of nonempty directed (usually) families of sets. In article, some set families connected with construction of ultrafilters (maximal filters) of MS majorizing a given a priory filter are treated as CAN. Shown, that in this case, under condition of the void intersection of all sets of the given filter, the resulting CAN variant is closed, but not canonically closed set for each of topologies Wallman and Stone types. This is connected with the fact established in the article that, for initial filter with property of the empty intersection of all its sets, the interior of generated by this filter AS is empty (at the same time, there are examples of control problems with opposite property: under empty intersection of sets for the family defining CAN, the interior of arising AS is not empty).
335-356