Wiesław Kotarski

Optimality conditions for n × n infinite-order parabolic coupled systems with control constraints and general performance index

A distributed control problem for n × n parabolic coupled systems involving operators with infinite order is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii-Milyutin theorem, the necessary and sufficient conditions of optimality are derived for the Dirichlet problem. Yet, the problem considered here is more general than the problems in El-Saify & Bahaa (2002, Optimal control for n × n hyperbolic systems involving operators of infinite order. Math. Slovaca, 52, 409-424), El-Zahaby (2002, Optimal control of systems governed by infinite order operators. Proceeding (Abstracts) of the International Conference of Mathematics (Trends and Developments) of the Egyptian Mathematical Society, Cairo, Egypt, 28-31 December 2002. J. Egypt. Math. Soc. (submitted)), Gali & El-Saify (1983, Control of system governed by infinite order equation of hyperbolic type. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 99-103), Gali et al. (1983, Distributed control of a system governed by Dirichlet and Neumann problems for elliptic equations of infinite order. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 83-87) and Kotarski et al. (200b, Optimal control problem for a hyperbolic system with mixed control-state constraints involving operator of infinite order. Int. J. Pure Appl. Math., 1, 241-254).

Optimal Control Problem for Infinite Order Hyperbolic System with Mixed Control-State Constraints

A distributed control problem for a hyperbolic system with mixed control-state constraints involving operator of infinite order is considered. The performance index is more general than the quadratic one and has an integral form. Making use of the Dubovitskii–Milyutin theorem,the optimality conditions for the Neumann problem are derived. Yet the problem considered here is more general than the problems in El-Zahaby [proceedings of the International conference on mathematics (Trends and Developments) of the Egyptian Mathematical Society, Cairo, Egypt, 28–31 December 2002, and Submitted for Publication in J Egyptian Math soc], Gali and El-Saify [proceedings of the International conference on functional–differential systems and related topics, vol III, Poland, 1983, pp 99–103] and Gali et al. [proceedings of the International conference on functional–differential systems and related topics, vol III, Poland, 1983, pp 83–87].

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

Polynomiography via Ishikawa and Mann Iterations

The aim of this paper is to present some modifications of the complex polynomial roots finding visualization process. In this paper Ishikawa and Mann iterations are used instead of the standard Picard iteration. The name polynomiography was introduced by Kalantari for that visualization process and the obtained images are called polynomiographs. Polynomiographs are interesting both from educational and artistic points of view. By the use of different iterations we obtain quite new polynomiographs that look aestheatically pleasing comparing to the ones from standard Picard iteration. As examples we present some polynomiographs for complex polynomial equation z 3 − 1 = 0, permutation and doubly stochastic matrices. We believe that the results of this paper can inspire those who may be interested in created automatically aesthetic patterns. They also can be used to increase functionality of the existing polynomiography software.

Optimality conditions for infinite order hyperbolic control problem with non-standard functional and time delay

A distributed control problem for the hyperbolic operator with infinite order and time delay is considered. The performance index has an integral form. Constraints on controls are imposed. To obtain optimality conditions for the Neumann problem the generalization of the Dubovitskii-Milyutin Theorem from [35, 36] was applied.

Automatic generation of aesthetic patterns with the use of dynamical systems

The aim of this paper is to present some modifications of the orbits generation algorithm of dynamical systems. The well-known Picard iteration is replaced by the more general one - Krasnosielskij iteration. Instead of one dynamical system, a set of them may be used. The orbits produced during the iteration process can be modified with the help of a probabilistic factor. By the use of aesthetic orbits generation of dynamical systems one can obtain unrepeatable collections of nicely looking patterns. Their geometry can be enriched by the use of the three colouring methods. The results of the paper can inspire graphic designers who may be interested in subtle aesthetic patterns created automatically.

On some specification of the Dubovicki-Milutin theorem for Pareto optimal problems

THE CLASSICAL Dubovicki-Milutin method described in [2] can be applied to obtain a necessary condition of optimality for optimization problems with only one equality constraint. In [9, 3, 6, 71 are given some generalizations of the Dubovicki-Milutin theorem which admit a greater number of equality constraints in optimization problems. Problems of Pareto optimality solved on the base of the Dubovicki-Milutin theorem have been considered in [ 11. By combination of the results of [9] and [l] in [4] is given a generalization of the Dubovicki-Milutin theorem for Pareto optimal problems with multi-equality constraints in a Banach space. In the present paper following [l] and [6] we derive some specification of the Dubovicki-Milutin theorem for Pareto optimal problems with multi-equality constraints given in the operator form.

Visualization of multidimensional data in explorative forecast

The aim of this paper is to present a new way of multidimensional data visualization for explorative forecast built for real meteorological data coming from the Institute of Meteorology and Water Management (IMGW) in Katowice, Poland. In the earlier works two first authors of the paper proposed a method that aggregates huge amount of data based on fuzzy numbers. Explorative forecast uses similarity of data describing situations in the past to those in the future. 2D and 3D visualizations of multidimensional data can be used to carry out its analysis to find hidden information that is not visible in the raw data e.g. intervals of fuzziness, fitting real number to a fuzzy number.

Polynomiography for the polynomial infinity norm via Kalantaris formula and nonstandard iterations

Survey of fixed point iteration processes.Extension of the pseudo-Newton method idea to other root finding methods.Modifications of the MMP-methods and their convergence visualizations.Non-trivial and intriguing polynomiographs have been obtained. In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-EulerSchrder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantaris recent results in finding the maximum modulus of a complex polynomial based on Newtons process with the Picard iteration to other MMP-processes with various non-standard iterations.

Using Genetic Algorithm to Aesthetic Patterns Design

This article presents possibilities of using a genetic algorithm as a method of artificial intelligence which is able to generate fractal structures with value of beauty. Fractal structure can be used as a utility model. The main problem is to define measure of beauty parameter. Using aesthetic measure algorithm can automatically, without human interaction, create nice looking various fractal structures.