Vladimir V. Karelin

Optimal Control Problems via Exact Penalty Functions

The nonsmoothness is viewed by many people as at least an undesirable (if not unavoidable) property. Our aim here is to show that recent developments in Nonsmooth Analysis (especially in Exact Penalization Theory) allow one to treat successfully even some quite ‘smooth’ problems by tools of Nonsmooth Analysis and Nondifferentiable Optimization. Our approach is illustrated by one Classical Control Problem of finding optimal parameters in a system described by ordinary differential equations.

Soil acidity adaptive control problem

Problem of soil acidity regularization is modeled as stochastic adaptive control problem with a linear difference equation of the dynamics of a field pH level. Stochastic component in the equation represents an individual time variability of soil acidity of an elementary section. We use Bayesian approach to determine a posteriori probability density function of the unknown parameters of the stochastic transition process. The Kullback–Leibler information divergence is used as a measure of difference between true distribution and its estimation. Algorithm for the construction of an adaptive stabilizing control in such a linear control system is proposed in the paper. Numerical realization of the algorithm is represented for a problem of a field soil acidity control.

Exact penalty methods for nonsmooth optimization

Methods of penalty functions are widely used in nonlinear programming. The idea of penalty function methods consists in reducing the problem of conditional optimization to a problem of the unconstrained optimization. Among the different approaches existing for such reduction we shall consider the method of exact penalty nondifferentiable functions. The implementation of exact penalty function methods first of all depends on the properties of an objective function and a function defining a constraint. Therefore various conditions are imposed on these functions to make it possible to solve our problem.

Minimax control in the singularly perturbed linear-quadratic stabilization problem

The output feedback stabilization problem is discussed. It is known that the lack of information about states does not permit to design a control which minimizes the quadratic functional for arbitrary initial states. In the paper, the minimax approach is considered and thereby the discrete minimax problem is solved. The main difference between the report and previous works is in the presence of regular and singular perturbations in the dynamics.

Optimal Control Problems and Penalization

The Exact Penalization Technique is applied to treat optimal control problems in a system described by ordinary differential equations. The resulting functional is essentially nonsmooth but directionally differentiable (even subdifferentiable). Differential equations are viewed as constraints and are “removed” by introducing an exact penalty function. The aim of the paper is to illustrate that well-known optimality conditions can be derived via Exact Penalty approach.

About constructing a dual polyhedral cone in R 3

The problem of constructing a dual cone of a convex polyhedral cone arises in many mathematical and applied researches. In the paper the method of constructing the dual cone to the acute convex polyhedral cone in R3 is proposed. The Householder transformation, creating a convex hull on the plane and projecting the point lying on the z-axis onto a corresponding face are the basic operations which we used.

A stochastic model of rumour spreading

In this paper, the new model for dynamics of spreading of rumour in a continuous time is suggested. Model depends on a parameter representing the probability of the outcome of interaction between the two spreaders. The process of spreading rumors is described by a system of linear differential equations.

Efficiency assessment of clinical decision support system

Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (грант № 14-01-31521 мол-а) и Санкт-Петербургского государственного университета (НИР, проект № 9.38.205.2014).

The allocation of resources between protection systems and redundancy modules

Recently, there are articles in which investigate problems associated with the detection of “attacks” on different systems, in particular Dos-attacks, or attacks like “denial of service” in the computer system and the subsequent changes in the input protocol of processing tasks. The problem of dynamic resource allocation between the protection of a system consisting of several modules of the same type, and the creation of new modules is considered. The most important role in the model plays a probability of a “failure” of a working system as a result of alleged “attack”.

About probabilistic model of the terminal operation

Suppose that similar terminals of several companies are in relative proximity to each other. Each company carries out transportation and storage of cargoes. If the terminal is crowded then the company can rent a portion of a terminal of another company. On contrary, if the terminal of some company is not sufficiently loaded, then the portion of the terminal can be suggested in the lease to another company.