Annals of the Rheumatic Diseases

The problem of modeling a solution is studied for a nonlinear system of differential equations with constant delay, inexactly known right-hand side, and inaccurately given initial state. The case is considered when the right-hand side of the system is a nonsmooth (it is only known that it is Lebesgue measurable) unbounded function (belonging to the space of square integrable functions in the Euclidean norm). An algorithm for solving this system that is stable to information noises and calculation errors is constructed. The algorithm is based on the concepts of feedback control theory. An estimate of the convergence rate of the algorithm is established. The possibility of using the algorithm to find an approximate solution to a system of ordinary differential equations is mentioned.

The paper is devoted to trajectory optimization for an inertial object moving in a plane with thrust bounded in magnitude in the presence of external forces. The aim is to maximize the longitudinal terminal velocity with the state constraint satisfied at each time to avoid a lateral collision with an obstacle. The linear tangent law is used as the basis for an algorithm that controls the direction of the thrust. Conditions for the existence of a solution are studied. Constraints on the initial lateral velocity and the time of the motion of the object are obtained. Since the linear tangent law violates the constraint for some motion times, a modified control law is proposed. A transcendental equation is obtained to find a critical value of time above which an undesired collision occurs. The corresponding conjecture is formulated, which allows us to eliminate the ambiguity that arises during the solution process. A method for solving the problem is presented and confirmed by numerical calculations.

A quadratic minimization problem is considered in Hilbert spaces under constraints given by a linear operator equation and a convex quadratic inequality. The main feature of the problem statement is that the practically available approximations to the exact linear operators specifying the criterion and the constraints converge to them only strongly pointwise rather than in the uniform operator norm, which makes it impossible to justify the use of the classical regularization methods. We propose a regularization method that is applicable in the presence of error estimates for approximate operators in pairs of other operator norms, which are weaker than the original ones. For each of the operators, the pair of corresponding weakened operator norms is obtained by strengthening the norm in the domain of the operator and weakening the norm in its range. The weakening of operator norms usually makes it possible to estimate errors in operators where this was fundamentally impossible in the original norms, for example, in the finite-dimensional approximation of a noncompact operator. From the original optimization formulation, a transition is made to the problem of finding a saddle point of the Lagrange function. The proposed numerical method for finding a saddle point is an iterative regularized extragradient two-stage procedure. At the first stage of each iteration, an approximation to the optimal value of the criterion is refined; at the second stage, the approximate solution with respect to the main variable is refined. Compared to the methods previously developed by the authors and working under similar information conditions, this method is preferable for practical implementation, since it does not require the gradient step size to converge to zero. The main result of the work is the proof of the strong convergence of the approximations generated by the method to one of the exact solutions to the original problem in the norm of the original space.

An optimal control problem is considered for a hybrid system in which continuous motion alternates with discrete changes (switchings) of the state space and control space. The switching times are determined as a result of minimizing a functional that takes into account the costs of each switching. Sufficient conditions for the optimality of such systems under additional constraints at the switching times are obtained. The application of the optimality conditions is demonstrated using academic examples.

In the framework of Baoding Liu’s uncertainty theory, some new concepts are introduced and their properties are considered. In particular, regular functions of uncertainty are introduced on an uncountable product of spaces. An analog of the Łomnicki–Ulam theorem from traditional probability theory is obtained. Necessary and sufficient conditions are specified under which a function defined on a Banach space of bounded functions is a distribution function for some uncertain mapping. Some notions of Liu’s theory are generalized for uncountably many objects. Examples showing the similarity and the difference between Liu’s theory and probability theory are analyzed. An application of Liu’s theory to estimation theory is considered with examples.

For a given multivalued mapping $$F:X\rightrightarrows Y$$ and a given element $$\tilde{y}\in Y$$ , the existence of a solution $$x\in X$$ to the inclusion $$F(x)\ni\tilde{y}$$ and its estimates are studied. The sets $$X$$ and  $$Y$$ are endowed with vector-valued metrics $$\mathcal{P}_{X}^{E_{+}}$$ and $$\mathcal{P}_{Y}^{M_{+}}$$ , whose values belong to cones $$E_{+}$$ and  $$M_{+}$$ of a Banach space  $$E$$ and a linear topological space $$M$$ , respectively. The inclusion is compared with a “model” equation $$f(t)=0$$ , where $$f:E_{+}\to M$$ . It is assumed that $$f$$ can be written as $$f(t)\equiv g(t,t)$$ , where the mapping $$g:{E}_{+}\times{E}_{+}\to M$$ orderly covers the set $$\{0\}\subset M$$ with respect to the first argument and is antitone with respect to the second argument and $$-g(0,0)\in M_{+}$$ . It is shown that, in this case, the equation $$f(t)=0$$ has a solution $$t^{*}\in E_{+}$$ . Further, conditions on the connection between $$f(0)$$ and $$F(x_{0})$$ and between the increments of $$f(t)$$ for $$t\in[0,t^{*}]$$ and the increments of $$F(x)$$ for all  $$x$$ in the ball of radius $$t^{*}$$ centered at $$x_{0}$$ for some $$x_{0}$$ are formulated, and it is shown that the inclusion has a solution in the ball under these conditions. The results on the operator inclusion obtained in the paper are applied to studying an integral inclusion.

A traditional object of study in the mathematical theory of optimal control is a controlled object with geometric constraints on the control vector  $$u$$ . At the same time, it turns out that sometimes it is more convenient to impose integral constraints on the control vector  $$u$$ . For example, in the theory of automatic design of optimal controllers, it is sometimes assumed that the control vector $$u$$ is not subject to any geometric constraints, but there is a requirement that the control $$u(t)$$ and its squared length $$|u(t)|^{2}$$ are Lebesgue summable on the corresponding interval. This circumstance, as well as the fact that the performance index has the form of a quadratic functional, makes it possible to construct an optimal control under rather broad assumptions. Quadratic integral constraints on controls can be interpreted as some energy constraints. Controlled objects under integral constraints on the controls are given quite a lot of attention in the mathematical literature on control theory. We mention the works of N.N. Krasovskii, E.B. Lee, L. Markus, A.B. Kurzhanskii, M.I. Gusev, I.V. Zykov, and their students. The paper studies a linear time-optimal problem, in which the terminal set is the origin, under an integral constraint on the control. Sufficient conditions are obtained under which the optimal time as a function of the initial state  $$x_{0}$$ is continuous.

A control reconstruction problem for dynamic deterministic affine-control systems is considered. This problem consists of constructing piecewise constant approximations of an unknown control generating an observed trajectory from discrete inaccurate measurements of this trajectory. It is assumed that the controls are constrained by known nonconvex geometric constraints. In this case, sliding modes may appear. To describe the impact of sliding modes on the dynamics of the system, the theory of generalized controls is used. The notion of normal control is introduced. It is a control that generates an observed trajectory and is defined uniquely. The aim of reconstruction is to find piecewise constant approximations of the normal control that satisfy given nonconvex geometric constraints. The convergence of approximations is understood in the sense of weak convergence in the space $$L^{2}$$ . A solution to the control reconstruction problem is proposed.

The problem of calculating points and magnitudes of discontinuities in the controls acting on a system described by a nonlinear vector ordinary differential equation is considered. A similar problem is well known in systems theory and belongs to the class of failure identification problems. This paper specifies a regularizing algorithm that solves the problem synchronously with the process of functioning of the control system. The algorithm is based on a feedback control method called the dynamic regularization method in the literature; this method was previously actively used in problems of online reconstruction of nonsmooth unknown disturbances. The algorithm described in this work is stable to information noises and computational errors.

We study the problem of optimal stimulation of demand based on a controlled version of Kaldor’s business cycle model. Using the approximation method, we prove a version of Pontryagin’s maximum principle in normal form containing an additional pointwise condition on the adjoint variable. The results obtained develop and strengthen the previous results in this direction.

The problem of guaranteed closed-loop guidance to a given set at a given time is studied for a linear dynamic control system described by differential equations with a fractional derivative of the Caputo type. The initial state is a priori unknown, but belongs to a given finite set. The information on the position of the system is received online in the form of an observation signal. The solvability of the guidance problem for the control system is analyzed using the method of Osipov–Kryazhimskii program packages. The paper provides a brief overview of the results that develop the program package method and use it in guidance problems for various classes of systems. This method allows us to connect the solvability condition of the guaranteed closed-loop guidance problem for an original system with the solvability condition of the open-loop guidance problem for a special extended system. Following the technique of the program package method, a criterion for the solvability of the considered guidance problem is derived for a fractional-order system. In the case where the problem is solvable, a special procedure for constructing a guiding program package is given. The developed technique for analyzing the guaranteed closed-loop guidance problem and constructing a guiding control for an unknown initial state is illustrated by the example of a specific linear mechanical control system with a Caputo fractional derivative.

The paper considers minimization problems with a free right endpoint on a given time interval for control affine systems of differential equations. For this class of problems, we study an estimate for the number of different zeros of switching functions that determine the form of the corresponding optimal controls. This study is based on analyzing nonautonomous linear systems of differential equations for switching functions and the corresponding auxiliary functions. Nonautonomous linear systems of third order are considered in detail. In these systems, the variables are changed so that the matrix of the system is transformed into a special upper triangular form. As a result, the number of zeros of the corresponding switching functions is estimated using the generalized Rolle’s theorem. In the case of a linear system of third order, this transformation is carried out using functions that satisfy a nonautonomous system of quadratic differential equations of the same order. The paper presents two approaches that ensure the extensibility of solutions of a nonautonomous system of quadratic differential equations to a given time interval. The first approach uses differential inequalities and Chaplygin’s comparison theorem. The second approach combines splitting a nonautonomous system of quadratic differential equations into subsystems of lower order and applying the quasi-positivity condition to these subsystems.

The issues of existence and uniqueness of a solution to the Cauchy problem are studied for a linear equation in a Banach space with a closed operator at the unknown function that is resolved with respect to a first-order integro-differential operator of Gerasimov type. The properties of resolving families of operators of the homogeneous equations are investigated. It is shown that sectoriality, i.e., belonging to the class of operators  $$\mathcal{A}_{K}$$ introduced here, is a necessary and sufficient condition for the existence of an analytical resolving family of operators in a sector. A theorem on the perturbation of operators of the class $$\mathcal{A}_{K}$$ is obtained, and two versions of the theorem on the existence and uniqueness of a solution to a linear inhomogeneous equation are proved. Abstract results are used to study initial–boundary value problems for an equation with the Prabhakar time derivative and for a system of partial differential equations with Gerasimov–Caputo time derivatives of different orders.

A subgroup $$H$$ of a group $$G$$ is called $$\mathbb{P}$$ -subnormal in $$G$$ whenever either $$H=G$$ or there is a chain of subgroups $$H=H_{0}\subset H_{1}\subset\mathinner{\ldotp\ldotp\ldotp}\subset H_{n}=G$$ such that $$|H_{i}:H_{i-1}|$$ is a prime for every $$i=1,2,\mathinner{\ldotp\ldotp\ldotp},n$$ . We study the structure of a finite group $$G$$ all of whose Schmidt subgroups are $$\mathbb{P}$$ -subnormal. The obtained results complement the answer to Problem 18.30 in the Kourovka Notebook..

A subgroup $$H$$ of a group $$G$$ is pronormal  if, for each $$g\in G$$ , the subgroups $$H$$ and $$H^{g}$$ are conjugate in $$\langle H,H^{g}\rangle$$ . Most of finite simple groups possess the following property  $$(\ast)$$ : each subgroup of odd index is pronormal in the group. The conjecture that all finite simple groups possess the property  $$(\ast)$$ was established in 2012 in a paper by E.P. Vdovin and the third author based on the analysis of the proof that Hall subgroups are pronormal in finite simple groups. However, the conjecture was disproved in 2016 by A.S. Kondrat’ev together with the second and third authors. In a series of papers by Kondrat’ev and the authors published from 2015 to 2020, the finite simple groups with the property  $$(\ast)$$ except finite simple linear and unitary groups with some constraints on natural arithmetic parameters were classified. In this paper, we construct series of examples of nonpronormal subgroups of odd index in finite simple linear and unitary groups over a field of odd characteristic, thereby making a step towards completing the classification of finite simple groups with the property  $$(\ast)$$ .