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On a finite time horizon, we consider a control system described by a vector differential equation with right-hand side that changes its structure at some times spaced by a distance that cannot be less than a certain given value. In between two adjacent structure change instants, the right-hand side is a function that is Lipschitz in state variables, continuous in time, and linear in the control and perturbation, which take values in some convex closed sets. It is assumed that at the structure change instants the solution of the system may experience a jump by a certain vector of which only the direction is known. A uniform mesh is specified on the system operation interval. The values of the state vector are measured (with an error) at the mesh points. We solve the problem of constructing an algorithm for the formation of a system control that ensures bringing the system trajectory to the minimum possible neighborhood of the goal set at the end time. A solution algorithm is indicated that is based on the constructions of positional control theory and is resistant to information interferences and computational errors.

Intersections of cones of index zero with spheres are investigated. Fields of the corresponding minimal manifolds are found. In particular, the cone $$\mathbb{K} = \left\{ {x_0^2 + x_1^2 + x_2^2 + x_3^2} \right\}$$ is considered. Its intersection with the sphere $$\mathbb{S}^{3}=\sum\nolimits_{i=0}^{3} x_{i}^{2}$$ is often called the Clifford torus $$\mathbb {T}$$, because Clifford was the first to notice that the metric of this torus as a submanifold of $$\mathbb {S}^3$$ with the metric induced from $$\mathbb {S}^3$$ is Euclidian. In addition, the torus $$\mathbb {T}$$ considered as a submanifold of $$\mathbb {S}^3$$ is a minimal surface. Similarly, it is possible to consider the cone $${\mathcal K} = \{ \sum\nolimits_{i = 0}^3x_0^2 = \sum _{i = 4}^7x_i^2\} $$, often called the Simons cone because he proved that $${\mathcal K}$$ specifies a single-valued nonsmooth globally defined minimal surface in ℝ8 which is not a plane. It appears that the intersection of $${\mathcal K}$$ with the sphere $$\mathbb{S}^7$$, like the Clifford torus, is a minimal submanifold of $$\mathbb{S}^7$$. These facts are proved by using the technique of quaternions and the Cayley algebra.

The minimality of surfaces defined by certain classical embeddings is proved and properties of sections of the surfaces are studied.

In this paper, a nonlinear distributed second order equation is considered. An algorithm for tracking a prescribed solution based on constructions of feedback control theory is designed. The algorithm is stable with respect to information noise and computation errors. It is designed for solutions in a sufficiently large time interval.

For a boundary controlled dynamic system, algorithms for solving the problem of tracking reference motion and the problem of tracking reference control are described. The algorithms are robust to information noise and computational errors. The solution method is based on the extremal shift method from the theory of positional differential games.

This paper is devoted to a specification of the method of open-loop control packages, a universal instrument for verification of the solvability of problems of closed-loop control for partially observable dynamical systems. Under the assumption that the control system and observed signal are linear and the set of the admissible initial states is finite, a structure of the corresponding open-loop control packages is specified and a finite-step backward construction is described, which provides a criterion for the solvability of a problem of guaranteed closed-loop guidance onto a target set at a prescribed time.

By higher-order elastics we mean solutions of the problem to minimize integrals of quadratic forms of curvatures of multidimensional curves. Differential equations of second-order elastics (minimizing a quadratic form of the curvature and torsion of the curve) are obtained. Surfaces minimizing integrals of the squared Gaussian curvature are found.

In the paper, the extension of a field of geodesics ℘ on a manifold N isometrically embedded in a Riemannian manifoldM is considered. The symplectic involute of the manifold N along the field ℘ is defined and a theorem is proved which gives a multidimensional analog of Jacobi’s envelope theorem.

For some classes of systems described by ordinary differential equations, a survey of algorithms for the dynamic reconstruction of inputs is presented. The algorithms described in the paper are stable with respect to information noises and computation errors; they are based on methods from the theory of ill-posed problems as well as on appropriate modifications of N.N. Krasovskii’s principle of extremal aiming, which is known in the theory of guaranteed control.

The paper is devoted to developing the method of program packages as a tool for investigating problems of positional control with incomplete information. The method is embedded in the field of guaranteed control theory and was stipulated by a number of constructions from this theory. Under the assumption that an a priori given set of initial positions of a controlled system is finite, it is established that the solvability of a guaranteed guidance problem in the class of program packages (or, which is the same, in the class of positional strategies) is equivalent to the solvability of this problem in the class of considerably simpler program operators, namely, in the class of idealized program packages.

For the boundary-controlled dynamic system obeying a parabolic differential equation with the Neumann boundary condition, the problems of following the reference motion, following the reference control, and guaranteed control (at domination of the controller resource) were solved on the basis of the N.N. Krasovskii method of extremal shift from the theory of positional differential games.

The paper introduces a geometric feature of separability of graphs for extremum equality-type boundary problems. To find an optimal value for a problem with an almost separable graph, the paper presents an iteration algorithm, each step of which minimizes Lagrangian function for the main variable with a fixed Lagrangian multiplier. This algorithm dates back to Krasovskii extremal shift method from differential game theory.

Problems of dynamical reconstruction of unknown distributed and boundary controls acting on nonlinear hyperbolic equations are discussed. Regularized algorithms reconstructing the unknown input during the evolution of a process are designed. In addition, algorithms for solving problems of robust boundary control are developed. These algorithms are stable with respect to informational noise and computational errors.

This paper deals with a specific inverse problem of dynamics for a system described by a parabolic inequality. The aim is to reconstruct the input (the control) of the system on the basis of an on-line measurement corrupted by an error. The techniques applied to the solution are a combination of those developed in positional control theory and the theory of ill-posed problems.

Problems of dynamical reconstruction of time-varying parameters of distributed systems are discussed. For several classes of systems described by hyperbolic equations algorithms for reconstruction of distributed inputs and intensities of point sources are described. These algorithms are dynamic (operate synchronously with systems' motions) and stable with respect to observation noises.