V. N. Ushakov

The approximation of reachable domains of control systems

Abstract The problem of constructing reachable domains (RDs) of a non-linear control system functioning over a finite time interval is considered. A method is proposed for the approximate construction of RDs, based on partitioning the phase space of the system by an ϵ-lattice. Estimate. are obtained for the accuracy of the approximate RDs. An example is presented.

Construction of a minimax solution for an eikonal-type equation

A formula for a minimax (generalized) solution of the Cauchy-Dirichlet problem for an eikonal-type equation is proved in the case of an isotropic medium providing that the edge set is closed; the boundary of the edge set can be nonsmooth. A technique of constructing a minimax solution is proposed that uses methods from the theory of singularities of differentiable mappings. The notion of a bisector, which is a representative of symmetry sets, is introduced. Special points of the set boundary—pseudovertices—are singled out and bisector branches corresponding to them are constructed; the solution suffers a “gradient catastrophe” on these branches. Having constructed the bisector, one can generate the evolution of wave fronts in smoothness domains of the generalized solution. The relation of the problem under consideration to one class of time-optimal dynamic control problems is shown. The efficiency of the developed approach is illustrated by examples of analytical and numerical construction of minimax solutions.

Approximate construction of attainability sets of control systems with integral constraints on the controls

Abstract The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.

The stability defect of sets in game control problems

The stability property in a game problem of the approach of a conflict-controlled system to a goal set at a fixed terminal moment is investigated. The notion of a stability defect is introduced for sets in the space of game positions.

On the representation of Fibonacci and Lucas numbers as the sum of three squares

We study the problem of expressing the Fibonacci numbers and the Lucas numbers as the sum of three squares of integers. We obtain the description of all numbers admitting such a representation.

On the stability property in a game-theoretic approach problem with fixed terminal time

We study the stability property in a game-theoretic problem on the approach of a conflict control system to an objective set at a fixed time.

On the coincidence of maximal stable bridges in two approach game problems for stationary control systems

Two approach game problems with a compact target set on a finite time interval are studied. The coincidence of their solutions is investigated.

Differential properties of integral funnels and stable bridges

Abstract Left derivatives of a multivalued mapping, similar to those previously introduced in /1, 2/, are applied to investigate the necessary and sufficient conditions for the integral funnel of a differential inclusion. In terms of the construction developed for analysing the value function of a differential game (see, e.g., /3, 4/) and generalized solutions of the Hamilton-Jacobi equation (viscosity solutions in the sense of /5, 6/), we describe the integral funnel using viscosity solutions of Bellmans equation. The properties of stable bridges are investigated using left derivatives. Necessary and sufficient conditions for stable bridges are stated ∗∗ .

On the coincidence of maximal stable bridges in two game problems

We consider a conflict control system on a finite time interval governed by an ordinary differential equation. More specifically, we study and compare two game problems describing the system approaching a terminal set M in the state space of the system [1]. In one of these problems, the first player uses a feedback control to ensure that the state vector of the system reaches M at the terminal time. In the other problem, a feedback control is used to ensure that the state vec� tor of the system reaches M no later than at the termi� nal time. In the approach proposed in [1], the central elements of the solving construction in both problems

Coincidence Criteria for Maximal Stable Bridges in Two Game Problems of Approach

We consider a finite-dimensional conflict-controlled system whose behavior on a finite time interval is described by a vector differential equation. We analyze two game problems of approach in the phase space. In both problems the same terminal set is considered: in the first case, one should guarantee that the phase vector of the system reaches the terminal set at the final instant of time; in the second case, the phase vector should reach the terminal set no later than the final time instant. It is natural to assume that the construction of a solution to the first problem is much simpler than the construction of a solution to the second problem; this fact is confirmed by available experience. The paper is devoted to finding conditions on the system and the terminal set under which the solutions to the above problems coincide. Using these conditions, one can replace the solution of the second problem by the simpler solution of the first problem.