LINEARIZED REQUIRED OPTIMALITY CONDITIONS IN ONE SMOOTH GURSA-DARBU SYSTEM MANAGEMENT PROBLEM

In class measurable (in Lebesgue sense) and bounded control vector functions, we consider one non-smooth optimal control problem of the Goursat – Darboux system with a multipoint quality functional, which is a generalization terminal type functional. Applying one modified version of the increment method, and assuming that the right side of the equation and the functional qualities in a vector state have derivatives in any direction, the necessary optimality condition in derivative terms in the direction flax general is proved. The case of a quasidifferentiable quality functional is considered. In particular, the minimax problem is studied. Under the assumption that the control region is convex, taking into account the properties of non-differentiable functions, the necessary optimality condition is established, which is an analog of the linearized integral principle of maximin, which is constructive in nature and generalizes the point wise linearized (differential) maximum principle.

Hirota Difference Equation and Darboux System: Mutual Symmetry

We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in R 3 . We demonstrated that specific properties of solutions of the HDE with respect to independent variables enabled introduction of an infinite set of discrete symmetries. We showed that degeneracy of the HDE with respect to parameters of these discrete symmetries led to the introduction of continuous symmetries by means of a specific limiting procedure. This enabled consideration of these symmetries on equal terms with the original HDE independent variables. In particular, the Darboux system appeared as an integrable equation where continuous symmetries of the HDE served as independent variables. We considered some cases of intermediate choice of independent variables, as well as the relation of these results with direct and inverse problems.

On a Discretization of Confocal Quadrics. A Geometric Approach to General Parametrizations

We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system may be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. Various sequences correspond to various discrete parametrizations. The coordinate functions of discrete confocal quadrics are computed explicitly. The theory is illustrated with a variety of examples in two and three dimensions. These include confocal coordinate systems parametrized in terms of Jacobi elliptic functions. Connections with incircular nets and a generalized Euler–Poisson–Darboux system are established.