Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T21:29:50.817Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic Evasion-Interrogation Games with Uncertainty in the Context of Electromagetics

Published online by Cambridge University Press:  28 May 2015

H. T. Banks ,
Shuhua Hu ,
K. Ito  and
Sarah Grove Muccio
H. T. Banks*
Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, USA
Shuhua Hu*
Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, USA
K. Ito*
Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, USA
Sarah Grove Muccio*
Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, USA
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

We consider two player electromagnetic evasion-pursuit games where each player must incorporate significant uncertainty into their design strategies to disguise their intension and confuse their opponent. In this paper, the evader is allowed to make dynamic changes to his strategies in response to the dynamic input with uncertainty from the interrogator. The problem is formulated in two different ways; one is based on the evolution of the probability density function of the intensity of reflected signal and leads to a controlled forward Kolmogorov or Fokker-Planck equation. The other formulation is based on the evolution of expected value of the intensity of reflected signal and leads to controlled backward Kolmogorov equations. In addition, a number of numerical results are presented to illustrate the usefulness of the proposed approach in exploring problems of control in a general dynamic game setting.

Keywords

35Q6183C5083C2265M32Electromagnetic evasion-pursuituncertaintyfeedback controlcomputational methodsKolmogorov and Fokker-Planck equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 3 , August 2011 , pp. 359 - 378
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Allen, L., An Introduction to Stochastic Processes with Applications to Biology, Pearson Education-Prentice Hall, New Jersy, 2003.Google Scholar
[2]Aubin, J.-P., Optima and Equilbria: An Introduction to Nonlinear Analysis, Spring-Verlag, Berlin Heidelberg, 1993.CrossRefGoogle Scholar
[3]Banks, H.T. and Bihari, K. L., Modeling and estimating uncertainty in parameter estimation, Inverse Problems, 17 (2001), 95–111.CrossRefGoogle Scholar
[4]Banks, H.T., Bortz, D., Pinter, G. A. and Potter, L. K., Modeling and imaging techniques with potential for application in bioterrorism, Chapter 6 in Bioterrorism: Mathematical Modeling Applications in Homeland Security, Banks, H.T. and Castillo-Chavez, C., eds., Frontiers in Applied Mathematics, SIAM, Philadelphia, 2003, 129–154.Google Scholar
[5]Banks, H.T. and Davis, J.L., A comparison of approximation methods for the estimation of probability distributions on parameters, Tech. Report CRSC-TR05-38, Center for Research in Scientific Computation, N.C. State University, Raleigh, NC, October, 2005; Applied Numerical Mathematics, 57 (2007), 753–777.Google Scholar
[6]Banks, H.T., Grove, S.L., Ito, K. and Toivanen, J.A., Static two-player evasion-interrogation games with uncertainty, CRSC-TR06-16, Center for Research in Scientific Computation, N.C. State University, Raleigh, NC, June, 2006; Comp. and Applied Math, 25 (2006), 289–306.Google Scholar
[7]Banks, H.T. and Ito, K., A unified framework for approximation in inverse problems for distributed parameter systems, Control–Theory and Advanced Technology, 4 (1988), 73–90.Google Scholar
[8]Banks, H.T. and Ito, K., Approximation in LQR problems for infinte dimensional systems with unbounded input operators, Journal of Mathematical Systems, Estimation, and Control, 7 (1997), 1–34.Google Scholar
[9]Banks, H.T., Ito, K., Kepler, G.M. and Toivanen, J.A., Material surface design to counter electromagnetic interrogation of targets, CRSC-TR04-37, Center for Research in Scientific Computation, N.C. State University, Raleigh, NC, November, 2004; SIAM J. Appl. Math., 66 (2006), 1027–1049.Google Scholar
[10]Banks, H.T., Ito, K. and Toivanen, J.A., Determination of interrogating frequencies to maximize electromagnetic backscatter from objects with material coatings, CRSC-TR05-30, Center for Research in Scientific Computation, N.C. State University, Raleigh, NC, August, 2005; Communications in Comp. Physics, 1 (2006), 357–377.Google Scholar
[11]Banks, H.T. and Kunisch, K., Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, 1989.CrossRefGoogle Scholar
[12]Banks, H.T. and Kurdila, A.J., Hysteretic control influence operators representing smart material actuators: Identification and approximation, CRSC-TR96-23, August, 1996; Proc. 35th IEEE Conf. on Decision and Control, (Kobe, Japan), December, 1996, 3711–3716.CrossRefGoogle Scholar
[13]Banks, H.T., Kurdila, A.J. and Webb, G., Identification of hysteretic control influence operators representing smart actuators, Part I: Formulation, CRSC-TR96-14, April 1996; Mathematical Problems in Engineering, 3 (1997), 287-328.CrossRefGoogle Scholar
[14]Banks, H.T., Kurdila, A.J. and Webb, G., Identification of hysteretic control influence operators representing smart actuators: Part II, Convergent approximations, CRSC-TR97-7, April, 1997; J. of Intelligent Material Systems and Structures, 8 (1997), 536–550.CrossRefGoogle Scholar
[15]Banks, H.T., Smith, R.C. and Wang, Y., Smart Material Structures: Modeling, Estimation, and Contol, Wiley, Paris, 1996.Google Scholar
[16]Berkovitz, L.D., A differential game with no pure strategy solution, Annals Math. Studies, No. 52, Princeton Univ. Press, Princeton, 1964, 175–194.Google Scholar
[17]Billingsley, P., Convergence of Probability Measures, John Wiley, New York, 1968.Google Scholar
[18]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93, Applied Mathematical Sciences, Springer-Verlag, Berlin, 2nd ed., 1998.Google Scholar
[19]Ethier, S.N. and Kurtz, T.G., Markov Processes: Characterization and Convergence, J. Wiley & Sons, New York, 1986.CrossRefGoogle Scholar
[20]Elliott, R.J., Kalton, N.J. and Markus, L., Saddle points for linear differential games, SIAM J. Control, 11 (1973), 100–112.CrossRefGoogle Scholar
[21]Filippov, A.F., On certain questions in the theory of optimal control, SIAM J. Control, 1 (1962), 76–84.Google Scholar
[22]Gard, T.C., Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988.Google Scholar
[23]Gardiner, C.W., Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
[24]Huber, P.J., Robust Statistics, Wiley & Sons, New York, 1981.CrossRefGoogle Scholar
[25]Jackson, J.D., Classical Electrodynamics, Wiley & Sons, New York, 1975.Google Scholar
[26]McShane, E.J., Necessary conditions in generalized-curve problems of the calculus of variations, Duke Mathematical Journal, 7, (1940), 1–27.CrossRefGoogle Scholar
[27]McShane, E.J., Generalized curves, Duke Math J., 6 (1940), 513–536.CrossRefGoogle Scholar
[28]McShane, E.J., Relaxed controls and variational problems, SIAM J. Control, 5 (1967), 438–485.CrossRefGoogle Scholar
[29]Oksendal, B., Stochastic Differentail Equations, 5th edition, Springer, Berlin, 2000.Google Scholar
[30]Neumann, J. von, Zur theorie der gesellschaftsspiele, Math. Ann., 100 (1928), 295–320.CrossRefGoogle Scholar
[31]Neumann, J. von and Morgenstern, O., Theory of Games and Economic Behavior, Princeton University Press, Princeton, NJ, 1944.Google Scholar
[32]Prohorov, Yu. V., Convergence of random processes and limit theorems in probability theory, Theor. Prob. Appl., 1 (1956), 157–214.CrossRefGoogle Scholar
[33]Warga, J., Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111–128.CrossRefGoogle Scholar
[34]Warga, J., Functions of relaxed controls, SIAM J. Control, 5 (1967), 628–641.CrossRefGoogle Scholar
[35]Warga, J., Optimal Control of Differential and Functional Equations, Academic Press, New York, NY, 1972.Google Scholar
[36]Wloka, J., Partial Differential Equations, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
[37]Young, L.C., Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. et Lettres, Varsovie, Cl. III, 30 (1937), 212–234.Google Scholar
[38]Young, L.C., Necessary conditions in the calculus of variations, Acta Math., 69 (1938), 239–258.CrossRefGoogle Scholar