The Solution Classical Feedback Optimal Control Problem for m-Persons Differential Game with Imperfect Information

Affiliation(s)

Department of Mathematics, Israel Institute of Technologies, Haifa, Israel.

All-Russian Research Institute for Opto-Physical Measurements, Moscow, Russia.

V.A. Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, Russia.

Department of Mathematics, Israel Institute of Technologies, Haifa, Israel.

All-Russian Research Institute for Opto-Physical Measurements, Moscow, Russia.

V.A. Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, Russia.

ABSTRACT

The paper presents a new approach to construct the Bellman function and optimal control directly by way of using strong large deviations principle for the solutions Colombeau-Ito’s SDE. The generic imperfect dynamic models of air-to-surface missiles are given in addition to the related simple guidance law. A four examples have been illustrated, corresponding numerical simulations have been illustrated and analyzed.

Cite this paper

J. Foukzon, E. Men’kova and A. Potapov, "The Solution Classical Feedback Optimal Control Problem for m-Persons Differential Game with Imperfect Information,"*Open Journal of Optimization*, Vol. 2 No. 1, 2013, pp. 16-25. doi: 10.4236/ojop.2013.21003.

J. Foukzon, E. Men’kova and A. Potapov, "The Solution Classical Feedback Optimal Control Problem for m-Persons Differential Game with Imperfect Information,"

References

[1] A. Lyasoff, “Path Integral Methods for Parabolic Partial Differential Equations with Examples from Computational Finance,” Mathematical Journal, Vol. 9, No. 2, 2004, pp. 399-422.

[2] D. Rajter-Ciric, “A Note on Fractional Derivatives of Colombeau Generalized Stochastic Processes,” Novi Sad Journal of Mathematics, Vol. 40, No. 1, 2010, pp. 111- 121.

[3] C. Martiasa, “Stochastic Integration on Generalized Function Spaces and Its Applications,” Stochastics and Stochastic Reports, Vol. 57, No. 3-4, 1996, pp. 289-301. doi:10.1080/17442509608834064

[4] M. Oberguggenberger and D. Rajter-Ciric, “Stochastic Differential Equations Driven by Generalized Positive Noise,” Publications de l’Institut Mathématique, Nouvelle Série, Vol. 77, No. 91, 2005, pp. 7-19.

[5] J. Foukzon, “The Solution Classical and Quantum Feedback Optimal Control Problem without the Bellman Equation,” 2009. http://arxiv.org/abs/0811.2170v4

[6] J. Foukzon, A. A. Potapov, “Homing Missile Guidance Law with Imperfect Measurements and Imperfect Information about the System,” 2012. http://arxiv.org/abs/1210.2933

[7] P. Bernhard and A.-L. Colomb, “Saddle Point Conditions for a Class of Stochastic Dynamical Games with Imperfect Information,” IEEE Transactions on Automatic Control, Vol. 33, No. 1, 1988, pp. 98-101. http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=367

[8] A. V. Kryazhimskii, “Differential Games of Approach in Conditions of Imperfect Information about the System,” Ukrainian Mathematical Journal, Vol. 27, No. 4, 1975, pp. 425-429. doi:10.1007/BF01085592

[9] J. F. Colombeau, “Elementary Introduction to New Generalized Functions,” North-Holland, Amsterdam, 1985.

[10] J. F. Colombeau, “New Generalized Functions and Multiplication of Distributions,” North-Holland, Amsterdam, 1984.

[11] H. Vernaeve, “Ideals in the Ring of Colombeau Generalized Numbers,” 2007. http://arxiv.org/abs/0707.0698

[12] E. Mayerhofer, “Spherical Completeness of the Non-Archimedian Ring of Colombeau Generalized Numbers,” Bulletin of the Institute of Mathematics Academia Sinica (New Series), Vol. 2, No. 3, 2007, pp. 769-783.

[13] J. Foukzon, “Large Deviations Principles of Non-Freidlin-Wentzell Type,” 2008. http://arxiv.org/abs/0803.2072

[14] S. Gutman, “On Optimal Guidance for Homing Missiles,” Journal of Guidance and Control, Vol. 2, No. 4, 1979, pp. 296-300. doi:10.2514/3.55878

[15] V. Glizer and V. Turetsky, “Complete Solution of a Differential Game with Linear Dynamics and Bounded Controls,” Applied Mathematics Research Express, Vol. 2008, 2008, p. 49. doi:10.1093/amrx/abm012

[16] M. Idan and T. Shima, “Integrated Sliding Model Autopilot-Guidance for Dual-Control Missiles,” Journal of Guidance, Control and Dynamics, Vol. 30, No. 4, 2007, pp. 1081-1089. doi:10.2514/1.24953

[17] W. Cai and J. Z. Wang, “Adaptive Wavelet Collocation Methods for Initial Value Boundary Problems of Nonlinear PDE’s,” Pentagon Reports, 1993. http://www.stormingmedia.us/44/4422/A442272.html10.1093/amrx/abm012

[1] A. Lyasoff, “Path Integral Methods for Parabolic Partial Differential Equations with Examples from Computational Finance,” Mathematical Journal, Vol. 9, No. 2, 2004, pp. 399-422.

[2] D. Rajter-Ciric, “A Note on Fractional Derivatives of Colombeau Generalized Stochastic Processes,” Novi Sad Journal of Mathematics, Vol. 40, No. 1, 2010, pp. 111- 121.

[3] C. Martiasa, “Stochastic Integration on Generalized Function Spaces and Its Applications,” Stochastics and Stochastic Reports, Vol. 57, No. 3-4, 1996, pp. 289-301. doi:10.1080/17442509608834064

[4] M. Oberguggenberger and D. Rajter-Ciric, “Stochastic Differential Equations Driven by Generalized Positive Noise,” Publications de l’Institut Mathématique, Nouvelle Série, Vol. 77, No. 91, 2005, pp. 7-19.

[5] J. Foukzon, “The Solution Classical and Quantum Feedback Optimal Control Problem without the Bellman Equation,” 2009. http://arxiv.org/abs/0811.2170v4

[6] J. Foukzon, A. A. Potapov, “Homing Missile Guidance Law with Imperfect Measurements and Imperfect Information about the System,” 2012. http://arxiv.org/abs/1210.2933

[7] P. Bernhard and A.-L. Colomb, “Saddle Point Conditions for a Class of Stochastic Dynamical Games with Imperfect Information,” IEEE Transactions on Automatic Control, Vol. 33, No. 1, 1988, pp. 98-101. http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=367

[8] A. V. Kryazhimskii, “Differential Games of Approach in Conditions of Imperfect Information about the System,” Ukrainian Mathematical Journal, Vol. 27, No. 4, 1975, pp. 425-429. doi:10.1007/BF01085592

[9] J. F. Colombeau, “Elementary Introduction to New Generalized Functions,” North-Holland, Amsterdam, 1985.

[10] J. F. Colombeau, “New Generalized Functions and Multiplication of Distributions,” North-Holland, Amsterdam, 1984.

[11] H. Vernaeve, “Ideals in the Ring of Colombeau Generalized Numbers,” 2007. http://arxiv.org/abs/0707.0698

[12] E. Mayerhofer, “Spherical Completeness of the Non-Archimedian Ring of Colombeau Generalized Numbers,” Bulletin of the Institute of Mathematics Academia Sinica (New Series), Vol. 2, No. 3, 2007, pp. 769-783.

[13] J. Foukzon, “Large Deviations Principles of Non-Freidlin-Wentzell Type,” 2008. http://arxiv.org/abs/0803.2072

[14] S. Gutman, “On Optimal Guidance for Homing Missiles,” Journal of Guidance and Control, Vol. 2, No. 4, 1979, pp. 296-300. doi:10.2514/3.55878

[15] V. Glizer and V. Turetsky, “Complete Solution of a Differential Game with Linear Dynamics and Bounded Controls,” Applied Mathematics Research Express, Vol. 2008, 2008, p. 49. doi:10.1093/amrx/abm012

[16] M. Idan and T. Shima, “Integrated Sliding Model Autopilot-Guidance for Dual-Control Missiles,” Journal of Guidance, Control and Dynamics, Vol. 30, No. 4, 2007, pp. 1081-1089. doi:10.2514/1.24953

[17] W. Cai and J. Z. Wang, “Adaptive Wavelet Collocation Methods for Initial Value Boundary Problems of Nonlinear PDE’s,” Pentagon Reports, 1993. http://www.stormingmedia.us/44/4422/A442272.html10.1093/amrx/abm012