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Vol 59, No 11 (2023)
Articles
On Nonlinear Boundary Value Problems for Differential Inclusions
Abstract
We consider autonomous differential inclusions with nonlinear boundary conditions. Sufficient conditions for the existence of solutions in the class of absolutely continuous functions are obtained for these inclusions. It is shown that the corresponding existence theorem applies to the Cauchy problem and the antiperiodic boundary value problem. The result is used to derive a new mean value inequality for continuously differentiable functions.
1443-1450
Behavior of Trajectories of a Four-Dimensional Model of HIV Infection
Abstract
A model of interaction between the human immunodeficiency virus and the human immune system is considered. Equilibria in the state space of the system and their stability are analyzed, and the ultimate bounds of the trajectories are constructed. It has been proved that the local asymptotic stability of the equilibrium corresponding to the absence of disease is equivalent to its global asymptotic stability. The loss of stability is shown to be caused by a transcritical bifurcation.
1451-1461
1462-1470
On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field
Abstract
We consider the first mixed problem for the Vlasov–Poisson system with an external magnetic field in a domain with piecewise smooth boundary. This problem describes the kinetics of a two-component high-temperature plasma under the influence of a self-consistent electric field and an external magnetic field. The existence of global weak solutions is proved. In the case of a cylindrical domain, sufficient conditions are obtained for the existence of global weak solutions with supports in a strictly internal cylinder; this corresponds to the confinement of high-temperature plasma in a mirror trap.
1471-1499
On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line
Abstract
We consider a special system of integral equations of convolution type with a monotone convex nonlinearity naturally arising when searching for stationary or limit states in various dynamic models of applied nature, for example, in models of the spread of epidemics, and prove theorems stating the existence or absence of a nontrivial bounded solution with limits at +@ depending on the values of these limits and on the structure of the matrix kernel of the system. We also study the uniqueness of such a solution assuming that it exists. Specific examples of systems whose parameters satisfy the conditions stated in our theorems are given.
1500-1514
On the Relationship Between the Pontryagin Maximum Principle and the Hamilton–Jacobi–Bellman Equation in Optimal Control Problems for Fractional-Order Systems
Abstract
We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is studied. It is proved that the costate variable in the Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of the optimal result functional calculated along the optimal motion.
1515-1521
On a Positional Control Problem for a Nonlinear Equation with Distributed Parameters
Abstract
We consider a guaranteed control problem for a nonlinear distributed equation of diffusion type. The problem is essentially to construct a feedback control algorithm ensuring that the solution of a given equation tracks the solution of a similar equation subjected to an unknown disturbance. The case in which a discontinuous unbounded function can be a feasible disturbance is studied. We solve the problem under conditions of inaccurate measurement of solutions of each of the equations at discrete instants of time and indicate a solution algorithm robust under information noise and calculation errors.
1522-1532
On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty
Abstract
We consider a linear-convex control system defined by a set of differential equations with continuous matrix coefficients. The system may have control parameters, as well as uncertainties (interference) the possible values of which are subject to strict pointwise constraints. For this system, over a finite period of time, taking into account the constraints, we study the problem of guaranteed hitting the target set from a given initial position despite the effect of uncertainty. The main stage of solving the problem is the construction of an alternating integral and a solvability set. To construct the latter, the greatest computational complexity is the calculation of the geometric difference between the target set and the set determined by the uncertainty. A two-dimensional example of this problem is considered for which a method is proposed for finding the solvability set without the need to calculate the convex hull of the difference of the support functions of the sets.
1533-1540
On the Construction of the Graph of Discrete States of a Switched Affine System
Abstract
The problem of constructing the graph of states of a switched affine system closed by a static state feedback is considered. To solve this problem, a constructive algorithm based on the study of the consistency of systems of linear algebraic inequalities is proposed.
1541-1549
Bounded Real Lemma for the Anisotropic Norm of Time-invariant Systems with Multiplicative Noises
Abstract
We consider a discrete-time-invariant system with multiplicative noise with implementation in the state space. The exogenous disturbance is chosen from the class of time-invariant ergodic sequences of nonzero colorness. We consider the level of mean anisotropy of the exogenous disturbance to be bounded by a known value. Conditions for the anisotropic norm to be bounded by a given number are obtained in terms of solving a matrix system of inequalities with a convex constraint of a special type. It is demonstrated how, on the basis of the obtained conditions, to construct a static state control that ensures the minimum value of the anisotropic norm of the system enclosed by this control.
1550-1560
On the Stability of Periodic Solutions of a Model Navier–Stokes Equation in a Thin Layer
Abstract
We study the existence and stability of periodic solutions of the model Navier–Stokes equation in a thin three-dimensional layer depending on the existence and stability of periodic solutions of a special limit two-dimensional equation.
1561-1565
Hopf Bifurcation in a Predator–Prey System with Infection
Abstract
We study a model of a predator–prey system with possible infection of prey in the form of a three-dimensional system of ordinary differential equations. Using the localization method of compact invariant sets, the existence of an attractor is proved and a compact positively invariant set is found that estimates its position. The conditions for the extinction of populations and the existence of equilibria are found. A numerical method for finding a Hopf bifurcation of the inner equilibrium is proposed and an example of an arising stable limit cycle is given.
1566-1570
On the Variation of the Nonlinearity Parameter in the “Super-Twisting” Algorithm
Abstract
We study the stability of a modified (with variation in the nonlinearity parameter) “super-twisting” algorithm. The analysis is based on majorizing the trajectories of the system with an arbitrary nonlinearity parameter by the trajectories of systems of the classical “super-twisting” algorithm. Stability conditions for the modified systems are obtained, as well as estimates for the size of the stability domain depending on system parameters
1571-1574
O seminare po kachestvennoy teorii differentsial'nykh uravneniy v Moskovskom universitete
Abstract
1575-1588