# American Institute of Mathematical Sciences

March  2017, 7(1): 1-20. doi: 10.3934/naco.2017001

 1 Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.B. 78, Karmiel 2161002, Israel 2 Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.Box 78, Karmiel 2161002, Israel 3 Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel

The first author is the corresponding author.

Received  September 2016 Published  February 2017

We consider an infinite horizon zero-sum linear-quadratic differential game in the case where the cost functional does not contain a control cost of the minimizing player (the minimizer). This feature means that the game under consideration is singular. For this game, novel definitions of the saddle-point equilibrium and game value are proposed. To obtain these saddle-point equilibrium and game value, we associate the singular game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. Using the solvability conditions, the solution of the cheap control game is reduced to solution of a Riccati matrix algebraic equation with an indefinite quadratic term. This equation is perturbed by a small parameter. Subject to a proper assumption, an asymptotic expansion of a stabilizing solution to this equation is constructed and justified. Using this asymptotic expansion, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.

Citation: Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
##### References:
 [1] F. Amato and A. Pironti, A note on singular zero-sum linear quadratic differential games, in Proceedings of the 33rd conference on decision and control Lake Buena Vista, FL, USA, (1994), 1533–1535. Google Scholar [2] B. D. O. Anderson, Y. Feng and W. Chen, A game theoretic algorithm to solve Riccati and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations in H∞ control in Optimization and Optimal Control, Theory and Applications (Eds. A. Chinchuluun, P. M. Pardalos, R. Enkhbat and I. Tseveendorj), Springer, New York, NY, USA, (2010), 277–308. Google Scholar [3] T. Basar and G. J. Olsder, Dynamic Noncooparative Game Theory SIAM Books, Philadelphia, PA, USA, 1999.  Google Scholar [4] D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems Academic Prees, New York, NY, USA, 1975.  Google Scholar [5] A. E. Bryson and Y. C. Ho, Applied Optimal Control Hemisphere, New York, NY, USA, 1975.  Google Scholar [6] K. Forouhar, S. J. Gibson and C. T. Leondes, Singular linear quadratic differential games with bounds on control, J. Optim. Theory Appl., 41 (1983), 341-348.  doi: 10.1007/BF00935229.  Google Scholar [7] K. Forouhar and C. T. Leondes, Singular differential game numerical techniques, J. Optim. Theory Appl., 37 (1982), 69-87.  doi: 10.1007/BF00934367.  Google Scholar [8] V. Y. Glizer, Asymptotic solution of zero-sum linear-quadratic differential game with cheap control for the minimizer, NoDEA Nonlinear Differential Equations Appl., 7 (2000), 231-258.  doi: 10.1007/s000300050006.  Google Scholar [9] V. Y. Glizer, Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence, SIAM J. Control Optim., 50 (2012), 2862-2888.  doi: 10.1137/110852784.  Google Scholar [10] V. Y. Glizer, Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays, in Variational and Optimal Control Problems on Unbounded Domains, Contemporary Mathematics Series, (Eds. G. Wolansky and A. J. Zaslavski), vol. 619, American Mathematical Society, Providence, RI, USA, (2014), 59–98. Google Scholar [11] V. Y. Glizer, Nash equilibrium in a singular two-person linear-quadratic differential game: a regularization approach, in Proceedings of 24th Mediterranean Conference on Control and Automation (MED 2016), Athens, Greece, (2016), 1041–1046. Google Scholar [12] V. Y. Glizer and O. Kelis, Solution of a zero-sum linear quadratic differential game with singular control cost of minimiser, J. Control Decis., 2 (2015), 155-184.   Google Scholar [13] V. Y. Glizer and O. Kelis, Solution of a singular infinite horizon zero-sum linear-quadratic differential game: a regularization approach, in Proceedings of the 23rd Mediterranean Con-ference on Control and Automation (MED2015), Torremolinos, Spain, (2015), 390–397. Google Scholar [14] V. Y. Glizer and O. Kelis, Singular infinite horizon quadratic control of linear systems with known disturbances: a regularization approach} Available online http://arxiv.org/abs/1603.01839, 2016. Google Scholar [15] V. Y. Glizer and J. Shinar, On the structure of a class of time-optimal trajectories,, Optimal Control Appl. Methods, 14 (1993), 271-279.  doi: 10.1002/oca.4660140405.  Google Scholar [16] V. Y. Glizer and V. Turetsky, A linear differential game with bounded controls and two information delays,, Optimal Control Appl. Methods, 30 (2009), 135-161.  doi: 10.1002/oca.850.  Google Scholar [17] R. Isaacs, Differential Games John Wiley and Sons, New York, NY, USA, 1967. Google Scholar [18] D. H. Jacobson, On values and strategies for infinite-time linear quadratic games,, IEEE Trans. Automat. Control, 22 (1977), 490-491.   Google Scholar [19] F. M. Hamelin and M. A. Lewis, A differential game theoretical analysis of mechanistic models for territoriality, J. Math. Biol., 61 (2010), 665-694.  doi: 10.1007/s00285-009-0316-1.  Google Scholar [20] Y. Hu, B. Oksendal and A. Sulem, Singular mean-field control games with applications to optimal harvesting and investment problems Avaibable online, https://arxiv.org/abs/1406.1863, 2014. Google Scholar [21] R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mex, Third Series, 5 (1960), 102-119.   Google Scholar [22] P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Academic Press, London, England, 1986.  Google Scholar [23] N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems Springer, New York, NY, USA, 1988. doi: 10.1007/978-1-4612-3716-7.  Google Scholar [24] G. A. Kurina, On a degenerate optimal control problem and singular perturbations, Soviet Math. Dokl., 18 (1977), 1452-1456.   Google Scholar [25] E. F. Mageirou, Values and strategies for infinite time linear quadratic games, IEEE Trans. Automat. Control, 21 (1976), 547-550.   Google Scholar [26] A. A. Melikyan, Singular characteristics of first order PDEs in optimal control and differential games,, J. Math. Sci., 103 (2001), 745-755.   Google Scholar [27] H. Mukaidani, A new design approach for solving linear quadratic Nash games of multiparameter singularly perturbed systems, IEEE Trans. Circuits Syst. I, Reg. Papers, 52 (2005), 960-974.  doi: 10.1109/TCSI.2005.846668.  Google Scholar [28] H. Mukaidani and V. Dragan, Control of deterministic and stochastic systems with several small parameters -a survey, Annals of the Academy of the Romanian Scientists, Series on Mathematics and its Applications, 1 (2009), 112-158.   Google Scholar [29] I. R. Petersen, Linear-quadratic differential games with cheap control, Systems Control Lett., 8 (1986), 181-188.  doi: 10.1016/0167-6911(86)90077-0.  Google Scholar [30] L. Schwartz, Analyse Mathematique vol. I, Hermann, Paris, France, 1967.  Google Scholar [31] J. Shinar, Solution techniques for realistic pursuit-evasion games, in Advances in Control and Dynamic Systems (Ed. C. Leondes), Academic Press, New York, NY, USA, (1981), 63–124. Google Scholar [32] J. Shinar and V. Y. Glizer, Application of receding horizon control strategy to pursuit-evasion problems, Optimal Control Appl. Methods, 16 (1995), 127-141.   Google Scholar [33] J. Shinar, V. Y. Glizer and V. Turetsky, Solution of a singular zero-sum linear-quadratic differential game by regularization, Int. Game Theory Rev., 16 (2014), 1-32.  doi: 10.1142/S0219198914400076.  Google Scholar [34] E. N. Simakova, Differential pursuit game, Autom. Remote Control, 28 (1967), 173-181.   Google Scholar [35] A. W. Starr and Y. C. Ho, Nonzero-sum differential games, J. Optim. Theory Appl., 3 (1969), 184-206.  doi: 10.1007/BF00929443.  Google Scholar [36] A. A. Stoorvogel, The singular zero-sum differential game with stability using $H_{∞}$ control theory, Math. Control Signals Systems, 4 (1991), 121-138.  doi: 10.1007/BF02551262.  Google Scholar [37] V. Turetsky and V. Y. Glizer, Robust state-feedback controllability of linear systems to a hyperplane in a class of bounded controls, J. Optim. Theory Appl., 123 (2004), 639-667.  doi: 10.1007/s10957-004-5727-y.  Google Scholar [38] V. Turetsky and V. Y. Glizer, Robust solution of a time-variable interception problem: a cheap control approach, Int. Game Theory Rev., 9 (2007), 637-655.  doi: 10.1142/S0219198907001631.  Google Scholar [39] V. Turetsky, V. Y. Glizer and J. Shinar, Robust trajectory tracking: differential game/cheap control approach, Internat. J. Systems Sci., 45 (2014), 2260-2274.  doi: 10.1080/00207721.2013.768305.  Google Scholar [40] H. Xu and K. Mizukami, Infinite-horizon differential games of singularly perturbed systems: a unified approach, Automatica J. IFAC, 33 (1997), 273-276.  doi: 10.1016/S0005-1098(96)00173-2.  Google Scholar [41] X. Wang and J. B. Cruz, Asymptotic ε-Nash equilibrium for 2nd order two-player nonzero-sum singular LQ games with decentralized control, in Proceedings of 17th IFAC World Congress, Seoul, Korea, (2008), 3970–3975. Google Scholar [42] Xing Wang, Chang-qi Tao and Guo-ji Tang, A class of differential quadratic programming problems, Appl. Math. Comput., 270 (2015), 369-377.  doi: 10.1016/j.amc.2015.08.041.  Google Scholar [43] Xing Wang, Ya-wei Qi, Chang-qi Tao and Yi-bin Xiao, A class of delay differential variational inequalities J. Optim. Theory Appl. Published online 31 August 2016, DOI 10.1007/s10957-016-1002-2. doi: 10.1007/s10957-016-1002-2.  Google Scholar

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##### References:
 [1] F. Amato and A. Pironti, A note on singular zero-sum linear quadratic differential games, in Proceedings of the 33rd conference on decision and control Lake Buena Vista, FL, USA, (1994), 1533–1535. Google Scholar [2] B. D. O. Anderson, Y. Feng and W. Chen, A game theoretic algorithm to solve Riccati and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations in H∞ control in Optimization and Optimal Control, Theory and Applications (Eds. A. Chinchuluun, P. M. Pardalos, R. Enkhbat and I. Tseveendorj), Springer, New York, NY, USA, (2010), 277–308. Google Scholar [3] T. Basar and G. J. Olsder, Dynamic Noncooparative Game Theory SIAM Books, Philadelphia, PA, USA, 1999.  Google Scholar [4] D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems Academic Prees, New York, NY, USA, 1975.  Google Scholar [5] A. E. Bryson and Y. C. Ho, Applied Optimal Control Hemisphere, New York, NY, USA, 1975.  Google Scholar [6] K. Forouhar, S. J. Gibson and C. T. Leondes, Singular linear quadratic differential games with bounds on control, J. Optim. Theory Appl., 41 (1983), 341-348.  doi: 10.1007/BF00935229.  Google Scholar [7] K. Forouhar and C. T. Leondes, Singular differential game numerical techniques, J. Optim. Theory Appl., 37 (1982), 69-87.  doi: 10.1007/BF00934367.  Google Scholar [8] V. Y. Glizer, Asymptotic solution of zero-sum linear-quadratic differential game with cheap control for the minimizer, NoDEA Nonlinear Differential Equations Appl., 7 (2000), 231-258.  doi: 10.1007/s000300050006.  Google Scholar [9] V. Y. Glizer, Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence, SIAM J. Control Optim., 50 (2012), 2862-2888.  doi: 10.1137/110852784.  Google Scholar [10] V. Y. Glizer, Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays, in Variational and Optimal Control Problems on Unbounded Domains, Contemporary Mathematics Series, (Eds. G. Wolansky and A. J. Zaslavski), vol. 619, American Mathematical Society, Providence, RI, USA, (2014), 59–98. Google Scholar [11] V. Y. Glizer, Nash equilibrium in a singular two-person linear-quadratic differential game: a regularization approach, in Proceedings of 24th Mediterranean Conference on Control and Automation (MED 2016), Athens, Greece, (2016), 1041–1046. Google Scholar [12] V. Y. Glizer and O. Kelis, Solution of a zero-sum linear quadratic differential game with singular control cost of minimiser, J. Control Decis., 2 (2015), 155-184.   Google Scholar [13] V. Y. Glizer and O. Kelis, Solution of a singular infinite horizon zero-sum linear-quadratic differential game: a regularization approach, in Proceedings of the 23rd Mediterranean Con-ference on Control and Automation (MED2015), Torremolinos, Spain, (2015), 390–397. Google Scholar [14] V. Y. Glizer and O. Kelis, Singular infinite horizon quadratic control of linear systems with known disturbances: a regularization approach} Available online http://arxiv.org/abs/1603.01839, 2016. Google Scholar [15] V. Y. Glizer and J. Shinar, On the structure of a class of time-optimal trajectories,, Optimal Control Appl. Methods, 14 (1993), 271-279.  doi: 10.1002/oca.4660140405.  Google Scholar [16] V. Y. Glizer and V. Turetsky, A linear differential game with bounded controls and two information delays,, Optimal Control Appl. Methods, 30 (2009), 135-161.  doi: 10.1002/oca.850.  Google Scholar [17] R. Isaacs, Differential Games John Wiley and Sons, New York, NY, USA, 1967. Google Scholar [18] D. H. Jacobson, On values and strategies for infinite-time linear quadratic games,, IEEE Trans. Automat. Control, 22 (1977), 490-491.   Google Scholar [19] F. M. Hamelin and M. A. Lewis, A differential game theoretical analysis of mechanistic models for territoriality, J. Math. Biol., 61 (2010), 665-694.  doi: 10.1007/s00285-009-0316-1.  Google Scholar [20] Y. Hu, B. Oksendal and A. Sulem, Singular mean-field control games with applications to optimal harvesting and investment problems Avaibable online, https://arxiv.org/abs/1406.1863, 2014. Google Scholar [21] R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mex, Third Series, 5 (1960), 102-119.   Google Scholar [22] P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Academic Press, London, England, 1986.  Google Scholar [23] N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems Springer, New York, NY, USA, 1988. doi: 10.1007/978-1-4612-3716-7.  Google Scholar [24] G. A. Kurina, On a degenerate optimal control problem and singular perturbations, Soviet Math. Dokl., 18 (1977), 1452-1456.   Google Scholar [25] E. F. Mageirou, Values and strategies for infinite time linear quadratic games, IEEE Trans. Automat. Control, 21 (1976), 547-550.   Google Scholar [26] A. A. Melikyan, Singular characteristics of first order PDEs in optimal control and differential games,, J. Math. Sci., 103 (2001), 745-755.   Google Scholar [27] H. Mukaidani, A new design approach for solving linear quadratic Nash games of multiparameter singularly perturbed systems, IEEE Trans. Circuits Syst. I, Reg. Papers, 52 (2005), 960-974.  doi: 10.1109/TCSI.2005.846668.  Google Scholar [28] H. Mukaidani and V. Dragan, Control of deterministic and stochastic systems with several small parameters -a survey, Annals of the Academy of the Romanian Scientists, Series on Mathematics and its Applications, 1 (2009), 112-158.   Google Scholar [29] I. R. Petersen, Linear-quadratic differential games with cheap control, Systems Control Lett., 8 (1986), 181-188.  doi: 10.1016/0167-6911(86)90077-0.  Google Scholar [30] L. Schwartz, Analyse Mathematique vol. I, Hermann, Paris, France, 1967.  Google Scholar [31] J. Shinar, Solution techniques for realistic pursuit-evasion games, in Advances in Control and Dynamic Systems (Ed. C. Leondes), Academic Press, New York, NY, USA, (1981), 63–124. Google Scholar [32] J. Shinar and V. Y. Glizer, Application of receding horizon control strategy to pursuit-evasion problems, Optimal Control Appl. Methods, 16 (1995), 127-141.   Google Scholar [33] J. Shinar, V. Y. Glizer and V. Turetsky, Solution of a singular zero-sum linear-quadratic differential game by regularization, Int. Game Theory Rev., 16 (2014), 1-32.  doi: 10.1142/S0219198914400076.  Google Scholar [34] E. N. Simakova, Differential pursuit game, Autom. Remote Control, 28 (1967), 173-181.   Google Scholar [35] A. W. Starr and Y. C. Ho, Nonzero-sum differential games, J. Optim. Theory Appl., 3 (1969), 184-206.  doi: 10.1007/BF00929443.  Google Scholar [36] A. A. Stoorvogel, The singular zero-sum differential game with stability using $H_{∞}$ control theory, Math. Control Signals Systems, 4 (1991), 121-138.  doi: 10.1007/BF02551262.  Google Scholar [37] V. Turetsky and V. Y. Glizer, Robust state-feedback controllability of linear systems to a hyperplane in a class of bounded controls, J. Optim. Theory Appl., 123 (2004), 639-667.  doi: 10.1007/s10957-004-5727-y.  Google Scholar [38] V. Turetsky and V. Y. Glizer, Robust solution of a time-variable interception problem: a cheap control approach, Int. Game Theory Rev., 9 (2007), 637-655.  doi: 10.1142/S0219198907001631.  Google Scholar [39] V. Turetsky, V. Y. Glizer and J. Shinar, Robust trajectory tracking: differential game/cheap control approach, Internat. J. Systems Sci., 45 (2014), 2260-2274.  doi: 10.1080/00207721.2013.768305.  Google Scholar [40] H. Xu and K. Mizukami, Infinite-horizon differential games of singularly perturbed systems: a unified approach, Automatica J. IFAC, 33 (1997), 273-276.  doi: 10.1016/S0005-1098(96)00173-2.  Google Scholar [41] X. Wang and J. B. Cruz, Asymptotic ε-Nash equilibrium for 2nd order two-player nonzero-sum singular LQ games with decentralized control, in Proceedings of 17th IFAC World Congress, Seoul, Korea, (2008), 3970–3975. Google Scholar [42] Xing Wang, Chang-qi Tao and Guo-ji Tang, A class of differential quadratic programming problems, Appl. Math. Comput., 270 (2015), 369-377.  doi: 10.1016/j.amc.2015.08.041.  Google Scholar [43] Xing Wang, Ya-wei Qi, Chang-qi Tao and Yi-bin Xiao, A class of delay differential variational inequalities J. Optim. Theory Appl. Published online 31 August 2016, DOI 10.1007/s10957-016-1002-2. doi: 10.1007/s10957-016-1002-2.  Google Scholar
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