Vol 292, No Suppl 1 (2016)

We consider an estimation problem for a backward stochastic differential equation in the presence of statistically uncertain noise. We use the approach of the theory of guaranteed estimation and assume that the statistically uncertain noise, as well as some processes entering the equation, is subject to integral constraints. In the linear case, we prove a theorem on the approximation of random information sets by deterministic sets as the diffusion coefficient vanishes. Examples are considered.

We consider the problem of the reachability of states that are elements of a topological space under constraints of asymptotic nature on the choice of an argument of a given objective mapping. We study constructions that have the sense of extensions of the original space and are implemented with the use of methods that are natural for applied mathematics but employ elements of extensions used in general topology. The study is oriented towards the application in the problem on the construction and investigation of properties of reachability sets for control systems.

Constructions involving an approximate observation of constraints in control problems, as well as various generalized regimes, were widely used by N.N. Krasovskii and his students. In particular, this approach was applied in the proof of N.N. Krasovskii and A.I. Subbotin’s fundamental theorem of the alternative, which made it possible to establish the existence of a saddle point in a nonlinear differential game. In the investigation of impulse control problems, Krasovskii used techniques from the theory of generalized functions, which formed the basis for many studies in this direction. A number of A.B. Kurzhanski’s papers are devoted to the solution of control problems related in one way or another to the construction of reachability sets. Control problems with incomplete information, duality issues for control and observation problems, and team control problems constitute a far from exhaustive list of research areas where Kurzhanskii obtained profound results. These studies are characterized by the use of a wide range of tools and methods from applied mathematics and various constructions as well as by the combination of theoretical investigations and procedures related to the possibility of computer modeling.

The research direction developed in the present paper mainly concerns the problem of constraint observation (including “asymptotic” constraints) and involves other issues. Nevertheless, the idea of constructing generalized elements of various nature (in particular, generalized controls) seems to be useful for the purpose of asymptotic analysis of control problems that do not possess stability as well as problems on the comparison of different tendencies in the choice of control in the form of dependences on a complex of factors inherent in the original real-life problem. The use of such tools as the Stone–Čech compactification and Wallman’s extension is, of course, oriented toward the study of qualitative issues. In the authors’ opinion, the combined application of the approaches to the construction of extensions used in control theory and in general topology holds promise from the point of view of both pure and applied mathematics. Apparently, the present paper can be considered as a certain step in this direction.

We consider estimation techniques for trajectory tubes of a nonlinear control system with uncertainty in the initial data and under the assumption of quadratic nonlinearity of the velocity vectors with respect to the states of the system. It is assumed that the uncertain initial states and admissible controls are subject to ellipsoidal constraints. We study problems of sensitivity of reachable sets and of their ellipsoidal estimates to a finite-dimensional parameter appearing in the constraints and in the dynamics of the uncertain control system. The results are based on algorithms and techniques of the theory of ellipsoidal estimation and the theory of differential inclusions.

We present first and second order conditions, both necessary and sufficient, for ≺-minimizers of vector-valued mappings over feasible sets with respect to a nontransitive preference relation ≺. Using an analytical representation of a preference relation ≺ in terms of a suitable family of sublinear functions, we reduce the vector optimization problem under study to a scalar inequality, from which, using the tools of variational analysis, we derive minimality conditions for the initial vector optimization problem.

The paper is devoted to the problem of approximating reachable sets of a nonlinear control system with state constraints given as a solution set of a nonlinear inequality. A state constraint elimination procedure based on the introduction of an auxiliary constraintfree control system is proposed. The equations of the auxiliary system depend on a small parameter. It is shown that the reachable set of the original system can be approximated in the Hausdorff metric by reachable sets of the auxiliary control system as the small parameter tends to zero. Estimates of the convergence rate are given.

We consider control synthesis problems for linear and bilinear differential systems. Two types of problems are studied: when controls are additive and when they enter the matrix of the system. For both types, we consider cases without uncertainty and cases with uncertainty, including additive parallelotope-valued uncertainties and interval uncertainties in the coefficients of the system. We continue to develop the methods of “polyhedral” synthesis of controls with the use of polyhedral (parallelotope-valued) solvability tubes. Nonlinear systems of ordinary differential equations describing the mentioned polyhedral solvability tubes are presented. We propose new control strategies, which can be calculated by explicit formulas based on the mentioned tubes. In addition, we consider similar synthesis problems for multistage systems.

We consider a control synthesis problem for nonlinear dynamic systems under parametric uncertainty and bounded measurement noises. Because of bounded disturbances in measurements of the state vector and the nonlinearity in the control object, the initially formulated control synthesis problem for a family of nonlinear systems as a generalized Zubov problem is transformed into a symbiosis of generalized Zubov–Bulgakov problems. The main result of the paper is the analytic solution of a minimax synthesis problem, which yields a constructive method for finding an invariant set.

We consider a feedback control problem for a system of ordinary differential equations in the case when only a part of coordinates of the phase vector are measured and propose a solution algorithm that is stable to perturbations. The algorithm is based on a combination of the theories of dynamical inversion and guaranteed control. It consists of two blocks: a block for the dynamical reconstruction of unmeasured coordinates and a control block.

A Dirichlet nonlinear problem for a second-order equation is considered on an interval. The problem is perturbed by a delta-like potential ε⁻¹Q(ε⁻¹x), where the function Q(ξ) is compactly supported and 0 < ε ≪ 1. A solution of this boundary value problem is constructed with accuracy up to O(ε) with the use of the method of matched asymptotic expansions. The obtained asymptotic approximation is validated by means of the fixed-point theorem. All types of boundary conditions are considered for a linear boundary value problem.

We consider an interpolation problem with minimum value of the uniform norm of the Laplace operator of interpolants for a class of bounded interpolated sequences. The data are interpolated at nodes of the grid formed by points from ℝ² with integer coordinates. Two-sided estimates for the uniform norm of the best interpolant are found, which improve known estimates.

A differential inclusion with values in a reflexive Banach space such that its righthand side is at each time a convex closed cone is considered. The form of the weak polar cone of the cone of strongly bounded solutions to the Cauchy problem for this inclusion is found. A solution is called strongly bounded if it is an absolutely continuous function (in a wide sense) and its derivative is essentially bounded.

We consider a game problem of approach for a system whose dynamics is described by a partial differential equation not of Kovalevskaya type, i.e., unsolved with respect to the time derivative. The equation with boundary conditions is written in a Hilbert function space in an abstract form as a differential operator equation. Using the method of resolving functionals, we obtain sufficient conditions for the approach of the system’s dynamical vector to a cylindrical terminal set. The results are exemplified by means of a model problem concerning a filtering process for a fluid in fractured porous rocks.