Stochastic Control edited by Chris Myers SCIYO Stochastic Control Edited by Chris Myers Published by Sciyo Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2010 Sciyo All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by Sciyo, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Jelena Marusic Technical Editor Goran Bajac Cover Designer Martina Sirotic Image Copyright Evv, 2010. Used under license from Shutterstock.com First published September 2010 Printed in India A free online edition of this book is available at www.sciyo.com Additional hard copies can be obtained from publication@sciyo.com Stochastic Control, Edited by Chris Myers p. cm. ISBN 978-953-307-121-3 SCIYO.COM WHERE KNOWLEDGE IS FREE free online editions of Sciyo Books, Journals and Videos can be found at www.sciyo.com Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Preface IX The Fokker-Planck equation 1 Shambhu N. Sharma and Hiren G. Patel The Itô calculus for a noisy dynamical system 21 Shambhu N. Sharma Application of coloured noise as a driving force in the stochastic differential equations 43 W.M.Charles Complexity and stochastic synchronization in coupled map lattices and cellular automata 59 Ricardo López-Ruiz and Juan R. Sánchez Zero-sum stopping game associated with threshold probability 81 Yoshio Ohtsubo Stochastic independence with respect to upper and lower conditional probabilities defined by Hausdorff outer and inner measures 87 Serena Doria Design and experimentation of a large scale distributed stochastic control algorithm applied to energy management problems 103 Xavier Warin and Stephane Vialle Exploring Statistical Processes with Mathematica7 125 Fred Spiring A learning algorithm based on PSO and L-M for parity problem 151 Guangyou Yang, Daode Zhang, and Xinyu Hu Improved State Estimation of Stochastic Systems via a New Technique of Invariant Embedding 167 Nicholas A. Nechval and Maris Purgailis Fuzzy identification of discrete time nonlinear stochastic systems 195 Ginalber L. O. Serra Contents VI Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Fuzzy frequency response for stochastic linear parameter varying dynamic systems 217 Carlos C. T. Ferreira and Ginalber L. O. Serra Delay-dependent exponential stability and filtering for time-delay stochastic systems with nonlinearities 235 Huaicheng Yan, Hao Zhang, Hongbo Shi and Max Q H. Meng Optimal filtering for linear states over polynomial observations 261 Joel Perez, Jose P. Perez and Rogelio Soto The stochastic matched filter and its applications to detection and de-noising 271 Philippe Courmontagne Wireless fading channel models: from classical to stochastic differential equations 299 Mohammed Olama, Seddik Djouadi and Charalambos Charalambous Information flow and causality quantification in discrete and continuous stochastic systems 329 X. San Liang Reduced-Order LQG Controller Design by Minimizing Information Loss 353 Suo Zhang and Hui Zhang The synthesis problem of the optimum control for nonlinear stochastic structures in the multistructural systems and methods of its solution 371 Sergey V. Sokolov Optimal design criteria for isolation devices in vibration control 393 Giuseppe Carlo Marano and Sara Sgobba Sensitivity analysis and stochastic modelling of the effective properties for reinforced elastomers 411 Marcin Kamiński and Bernd Lauke Stochastic improvement of structural design 437 Soprano Alessandro and Caputo Francesco Modelling earthquake ground motions by stochastic method 475 Nelson Lam, John Wilson and Hing Ho Tsang VII Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 Quasi-self-similarity for laser-plasma interactions modelled with fuzzy scaling and genetic algorithms 493 Danilo Rastovic Efficient Stochastic Simulation to Analyze Targeted Properties of Biological Systems 505 Hiroyuki Kuwahara, Curtis Madsen, Ivan Mura, Chris Myers, Abiezer Tejeda and Chris Winstead Stochastic Decision Support Models and Optimal Stopping Rules in a New Product Lifetime Testing 533 Nicholas A. Nechval and Maris Purgailis A non-linear double stochastic model of return in financial markets 559 Vygintas Gontis, Julius Ruseckas and Aleksejus Kononovičius Mean-variance hedging under partial information 581 M. Mania, R. Tevzadze and T. Toronjadze Pertinence and information needs of different subjects on markets and appropriate operative (tactical or strategic) stochastic control approaches 609 Vladimir Šimović and Vladimir Šimović, j.r. Fractional bioeconomic systems: optimal control problems, theory and applications 629 Darya V. Filatova, Marek Grzywaczewski and Nikolai P. Osmolovskii Uncertainty presents signicant challenges in the reasoning about and controlling of complex dynamical systems. To address this challenge, numerous researchers are developing improved methods for stochastic analysis. This book presents a diverse collection of some of the latest research in this important area. In particular, this book gives an overview of some of the theoretical methods and tools for stochastic analysis, and it presents the applications of these methods to problems in systems theory, science, and economics. The rst section of the book presents theoretical methods and tools for the analysis of stochastic systems. The rst two chapters by Sharma et al. present the Fokker-Planck equation and the Ito calculus. In Chapter 3, Charles presents the use of colored noise with stochastic differential equations. In Chapter 4, Lopez-Ruiz and Sanchez discuss coupled map lattices and cellular automata. In Chapter 5, Ohtsubo presents a game theoretic approach. In Chapter 6, Doria presents an approach that uses Hausdorff outer and inner measures. In Chapter 7, Warin and Vialle for analysis using distributed algorithms. Finally, in Chapter 8, Spiring explores the use of Mathematica7. The second section of the book presents the application of stochastic methods in systems theory. In Chapter 9, Yang et al. present a learning algorithm for the parity problem. In Chapter 10, Nechval and Pugailis present an improved technique for state estimation. In Chapter 11, Serra presents a fuzzy identication method. In Chapter 12, Ferreira and Serra present an application of fuzzy methods to dynamic systems. The next three chapters by Yan et al., Perez et al., and Courmontagne explore the problem of ltering for stochastic systems. In Chapter 16, Olama et al. look at wireless fading channel models. In Chapter 17, Liang considers information ow and causality quantication. The last two chapters of this section by Zhang and Zhang and Sokolov consider control systems. The third section of the book presents the application of stochastic methods to problems in science. In Chapter 20, Marano and Sgobba present design criteria for vibration control. In Chapter 21, Kaminski and Lauke consider reinforced elastomers. In Chapter 22, Alessandro and Francesco discuss structural design. In Chapter 23, Lam et al. apply stochastic methods to the modeling of earthquake ground motion. In Chapter 24, Rastovic addresses laser-plasma interactions. Finally, in Chapter 25, Kuwahara et al. apply new, efcient stochastic simulation methods to biological systems. Preface X The nal section of the book presents the application of stochastic methods to problems in economics. In Chapter 26, Nechval and Purgailis consider the problem of determining a products lifetime. In Chapter 27, Gontis et al. applies a stochastic model to nancial markets. In Chapter 28, Mania et al. take on the problem of hedging in the market. In Chapter 29, Simovic and Simovic apply stochastic control approaches to tactical and strategic operations in the market. Finally, in Chapter 30, Darya et al. consider optimal control problems in fractional bio-economic systems. Editor Chris Myers University of Utah U.S.A. [...]... 1 i 2 ,1 j 2 and A1 x 2 A2 x1 x 2 ax13 bx 2 x12 (3ax1 bx 2 ) P 11 (2bx1 ) P12 , B 11 2 P12 , 2 B12 B 21 P 11 ( 3ax12 2bx1 x 2 ) P12 ( bx12 ) P22 3aP 11 3bP 11 P12 , 2 B22 2 P12 ( 3ax12 2bx1 x 2 ) 2( bx12 ) P22 6aP 11 P12 4bP12 2bP 11 P22 1 2 2 B x12 n (2n)(2n 1) B P 11 x12 n 2 2 Evolution equations... x12 x14 1 2 E ( x1 , x 2 ) x 2 a 2 2 4 (30) From equations (13 ), (28), and (30), we obtain 1 2 dE ( x1 , x 2 ) ((x1 ax13 ) x 2 x 2 (x1 x 2 ax13 bx 2 x12 ) B x12 n )dt 2 B x1n dwt After a simple calculation, we have the following SDE: 1 2 1 2 2 dE ( x 2 bx 2 x12 B x12 n u2 )dt B x1n dwt u dvt 2 2 The qualitative analysis of the stochastic problem... differential equations (Arnold 19 95), 1 F1 (t , x1 , x1 , x 2 , x 2 ), x 2 F2 (t , x1 , x1 , x 2 , x 2 ), x after introducing the noise processes along the components equations, the above can be re-written as ( x1 , x 2 ) of 1 F1 (t , x1 , x1 , x 2 , x 2 , B1 ), x 2 F2 (t , x1 , x1 , x 2 , x 2 , B2 ) x the coupled (1) (2) Equations (1) -(2) constitute a system of two... n 1 , z 2 , z 1 , z 0 r , ) can expectation of the conditional p ( z n , z n 1 , z 2 , z 1 , z 0 x n , x n 1 , x n 2 , x 2 , x 1 , x 0 ), i.e be regarded as probability density 26 Stochastic Control p ( z n , z n 1 , , z 2 , z 1 , z 0 r , ) E ( p ( z n , z n 1 , , z 2 , z 1 , z 0 x n , x n 1 , x n 2 , x 2 , x 1 , x 0 ) r , ), where x k 1 ... the stochastic system of this chapter, i.e p p p p p bx12 p p ax13 x 2 x1 x2 x 2 x 2 x 2 x1 t 2n2 2 p B (2n)(2n 1) x 1 bx x x 2 2 2 2 1 p Alternatively, the stochastic differential system can be analysed qualitatively involving the Itô differential rule, see equation (13 ) of the chapter The energy function for the stochastic system of this chapter is x12 x14 1 2... approach for a stochastic Duffing-van der Pol system, Differential Equations and Dynamical Systems (An international Journal for theory, applications and computer simulations), 16 (4), 3 513 77 DOI: 10 .10 07/s125 91- 008-0 019 -x http://www.springerlink.com/content/t5 315 t2k6 215 415 1/ The Itô calculus for a noisy dynamical system 21 2 X The Itô calculus for a noisy dynamical system Shambhu N Sharma Department of... x1 , x3 ) qt ( x1 , x 2 )q s ( x 2 , x3 )dx 2 Consider the multi-dimensional probability density p ( x1 , x 2 ) p ( x1 x 2 ) p ( x 2 ) and integrating over the variable x 2 , we have p( x1 ) p( x1 x 2 ) p( x 2 )dx 2 , or where p( x1 ) qt1 ,t2 ( x1 , x 2 ) p( x 2 )dx 2 , qt1 ,t2 ( x1 , x 2 ) is probability density q t the transition probability density and 1 ,t 2 function Ee iu ( xt1... probability density and 1 ,t 2 function Ee iu ( xt1 xt 2 ) , ( x1 , x 2 ) (1) t1 t 2 The transition is the inverse Fourier transform of the characteristic i.e qt1 ,t2 ( x1 , x 2 ) 1 iu ( xt xt ) iu ( x x ) e 1 2 Ee 1 2 du 2 (2) Equation (1) in combination with equation (2) leads to p ( x1 ) 1 iu ( xt xt ) iu ( x x ) e 1 2 (Ee 1 2 ) p( x2 )dx2 du 2 (3) The characteristic function is... multi-dimensional stochastic differential equation Choose and x1 x3 , x2 x4 , x3 F1 (t , x1 , x 2 , x3 , x 4 , B1 ), x F (t , x , x , x , x , B ) 4 2 1 2 3 4 2 By considering a special case of the above system of equations, we have dx1 x3 dt , dx 2 x 4 dt , 24 and Stochastic Control dx3 f 3 (t , x1 , x 2 , x3 , x 4 )dt g 3 (t , x1 , x 2 , x3 , x 4 )dB1 , dx3 f 4 (t , x1 , x 2 ,... iu ( xt1 xt1 ) (iu ) n ( xt1 xt2 ) n n! 0 n After introducing the definition of the characteristic function, equation (3) can be recast as 1 (iu ) n e iu ( x1 x2 ) ( ( xt1 xt2 ) n ) p( x 2 )dx 2 du 2 n! 0 n 1 1 ( (iu ) n e iu ( x1 x2 ) du ) ( xt1 xt2 ) n p( x 2 )dx 2 0 n n! 2 p( x1 ) The Fokker-Planck equation The term density 5 1 n iu ( x x ) n e 1 2 (iu . time nonlinear stochastic systems 19 5 Ginalber L. O. Serra Contents VI Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter. , )( 21 tt xxiu Ee i.e. . 2 1 ),( )( )( 21, 212 1 21 duEeexxq tt xxiu xxiu tt (2) Equation (1) in combination with equation (2) leads to .)()( 2 1 )( 22 )( )( 1 212 1 dudxxpEeexp tt xxiu xxiu . density )()(),( 2 212 1 xpxxpxxp and integrating over the variable 2 x , we have ,)()()( 22 211 dxxpxxpxp or ,)(),()( 22 21, 1 21 dxxpxxqxp tt (1) where ),( 21, 21 xxq tt is the