Proceedings of the Steklov Institute of Mathematics最新文献

For a given multivalued mapping (F:Xrightrightarrows Y) and a given element (tilde{y}in Y), the existence of a solution (xin X) to the inclusion (F(x)nitilde{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 (tin[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.

We consider a regularization of the Lagrange multiplier rule (LMR) in the nondifferential form in a convex constrained optimization problem with an operator equality constraint in a Hilbert space and a finite number of functional inequality constraints. The objective functional of the problem is assumed to be strongly convex, and the convex closed set of its admissible elements also belongs to a Hilbert space. The constraints of the problem contain additively included parameters, which enables using the so-called perturbation method to study it. The main purpose of the regularized LMR is the stable generation of generalized minimizing sequences (GMSs), which approximate the exact solution of the problem using extremals of the regular Lagrange functional. The regularized LMR itself can be interpreted as a GMS-generating (regularizing) operator, which assigns to each set of input data of the constrained optimization problem the extremal of its corresponding regular Lagrange functional, in which the dual variable is generated following one or another procedure for stabilizing the dual problem. The main attention is paid to (1) studying the connection between the dual regularization procedure and the subdifferential properties of the value function of the original problem; (2) proving the convergence of this procedure in the case of solvability of the dual problem; (3) an appropriate update of the regularized LMR; (4) obtaining the classical LMR as a limiting version of its regularized analog.

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.

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 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.

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}subsetmathinner{ldotpldotpldotp}subset H_{n}=G)such that (|H_{i}:H_{i-1}|) is a prime for every (i=1,2,mathinner{ldotpldotpldotp},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..

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 toinformation 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 toa system of ordinary differential equations is mentioned.

For conflict-controlled dynamical systems satisfying the conditions of generalized uniqueness and uniform boundedness, the solvability of the minimax problem in the class of relaxed controls is studied. The issues of properness of such a relaxation are considered; i.e., the possibility of approximating relaxed controls in the space of strategic measures by embeddings of ordinary controls is analyzed. For this purpose, the dependence of the set of measures on the general marginal distribution specified on one of the factors of the base space is studied. The continuity of this dependence in the Hausdorff metric defined by the metric corresponding to the (ast)-weak topology in the space of measures is established. The density of embeddings of ordinary controls and control–disturbance pairs in sets of corresponding relaxed controls in the (ast)-weak topologies is also shown.

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.

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.