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doi: 10.3934/jimo.2021034
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## Linear-quadratic optimal control for discrete-time stochastic descriptor systems

 1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China 2 School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China

* Corresponding author

Received  October 2020 Revised  December 2020 Early access February 2021

In this paper, an optimal control model ruled by a class of linear discrete-time stochastic descriptor systems is considered under quadratic index performance. Employing dynamic programming method, a recurrence equation to simplify the optimal control problem is presented provided that the descriptor systems are both regular and impulse-free. When the objective function is quadratic, according to the recurrence equation, a discrete-time linear-quadratic optimal control problem is completely settled, that is, optimal controls and optimal values of the problem are both obtained through analytical expressions. At last, a numerical example about linear-quadratic optimal control for a discrete-time stochastic descriptor system is provided to illustrate the validness of the results derived.

Citation: Yadong Shu, Bo Li. Linear-quadratic optimal control for discrete-time stochastic descriptor systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021034
##### References:
 [1] K. Avrachenkov, O. Habachi, A. Piunovskiy and Y. Zhang, Infinite horizon optimal impulsive control with applications to Internet congestion control, International Journal of Control, 88 (2015), 703-716.  doi: 10.1080/00207179.2014.971436.  Google Scholar [2] A. V. Balakrishnan, On stochastic bang-bang control, Applied Mathematics and Optimization, 6 (1980), 91-96.  doi: 10.1007/BF01442885.  Google Scholar [3] M. Basin and and J. Rodriguez-Gonzalez, Optimal control for linear systems with time delay in control input, IEEE Transactions on Automatic Control, 51 (2006), 91-97.  doi: 10.1109/TAC.2005.861718.  Google Scholar [4] A. Cairns, Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time, Astin Bulletin, 30 (2000), 19-55.  doi: 10.2143/AST.30.1.504625.  Google Scholar [5] S. L. Campbell, Singular Systems of Differential Equations II, Pitman, Boston, Mass.-London, 1982.  Google Scholar [6] M. Chadli, H. R. Karimi and P. Shi, On stability and stabilization of singular uncertain Takagi-Sugeno fuzzy systems, Journal of the Franklin Institute, 351 (2014), 1453-1463.  doi: 10.1016/j.jfranklin.2013.11.008.  Google Scholar [7] L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica J. IFAC, 46 (2010), 1074-1080.  doi: 10.1016/j.automatica.2010.03.005.  Google Scholar [8] D. Cobb, Controllability, observability, and duality in singular systems, IEEE Transactions on Automatic Control, 29 (1984), 1076-1082.  doi: 10.1109/TAC.1984.1103451.  Google Scholar [9] L. Dai, Singular Control Systems, Springer-Verlag, Berlin, Germany, 1989. doi: 10.1007/BFb0002475.  Google Scholar [10] J.-e Feng, P. Cui and Z. Hou, Singular linear quadratic optimal control for singular stochastic discrete-time systems, Optimal Control Applications and Methods, 34 (2013), 505-516.  doi: 10.1002/oca.2033.  Google Scholar [11] J. Y. Ishihara and M. H. Terra, On the Lyapunov theorem for singular systems, IEEE Transactions on Automatic Control, 47 (2002), 1926-1930.  doi: 10.1109/TAC.2002.804463.  Google Scholar [12] R. E. Kalman, Contribution to the theory of optimal control, Boletin Sociedad Matem$\acute{a}$tica Mexicana, 5 (1960), 102–119.  Google Scholar [13] Y. Kang and Y. Zhu, Bang-bang optimal control for multi-stage uncertain systems, International Journal on Information, 15 (2012), 3229-3237.   Google Scholar [14] D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2012. Google Scholar [15] N. Kumaresan and P. Balasubramaniam, Optimal control for stochastic nonlinear singular system using neural networks, Computers and Mathematics with Applications, 56 (2008), 2145-2154.  doi: 10.1016/j.camwa.2008.03.041.  Google Scholar [16] P. Kunkel and V. Mehrmann, The linear quadratic optimal control problem for linear descriptor systems with variable coefficients, Mathematics of Control, Signals, and Systems, 10 (1997), 247-264.  doi: 10.1007/BF01211506.  Google Scholar [17] B. Li and and Y. Zhu, The piecewise parametric optimal control of uncertain linear quadratic models, International Journal of Systems Science, 50 (2019), 961-969.  doi: 10.1080/00207721.2019.1586003.  Google Scholar [18] F. Liao, Z. Ren, M. Tomizuka and J. Wu, Preview control for impulse-free continuous-time descriptor systems, International Journal of Control, 88 (2015), 1142-1149.  doi: 10.1080/00207179.2014.996769.  Google Scholar [19] J. Lian, C. Li and D. Liu, Input-to-state stability for discrete-time nonlinear switched singular systems, IET Control Theory Appl., 11 (2017), 2893–2899. doi: 10.1049/iet-cta.2017.0028.  Google Scholar [20] W. Liu, X. Liang, Y. Ma and W. Liu, Aircraft trajectory optimization for collision avoidance using stochastic optimal control, Asian Journal of Control, 21 (2019), 2308-2320.  doi: 10.1002/asjc.1855.  Google Scholar [21] D. G. Luenberger, Dynamic equations in descriptor form, IEEE Transactions on Automatic Control, 22 (1977), 312-321.  doi: 10.1109/tac.1977.1101502.  Google Scholar [22] D. G. Luenberger and A. Arbel, Singular dynamic Leontief systems, Econometrica, 45 (1977), 991-995.  doi: 10.2307/1912686.  Google Scholar [23] G. P. Mantas and N. J. Krikelis, LQ optimal control for discrete descriptor systems with a complete set of boundary information, International Journal of Control, 47 (1988), 1467-1477.  doi: 10.1080/00207178808906108.  Google Scholar [24] D. S. Naidu, Singular perturbations and time scales in control theory and applications: An overview., Dynamics of Continuous Discrete & Impulsive Systems Ser. B Appl. Algorithms, 9 (2002), 233-278.   Google Scholar [25] H. Peng, Fei. Li, J. Liu and Z. Ju, A symplectic instantaneous optimal control for robot trajectory tracking with differential-algebraic equation models, IEEE Transactions on Industrial Electronics, 67 (2020), 3819-3829.  doi: 10.1109/TIE.2019.2916390.  Google Scholar [26] Y. Shu and Y. Zhu, Optimistic value based optimal control for uncertain linear singular systems and application to a dynamic input-output model, ISA Transactions, 71 (2017), 235-251.  doi: 10.1016/j.isatra.2017.08.007.  Google Scholar [27] Y. Shu and Y. Zhu, Stability analysis of uncertain singular systems, Soft Computing, 22 (2018), 5671-5681.  doi: 10.1007/s00500-017-2599-2.  Google Scholar [28] K. Wang, L. Xie and Y. Liu, Characteristics of self-excited oscillation in a curved diffuser., Journal of Propulsion Technology, 39 (2018), 1955-1964.   Google Scholar [29] W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM Journal on Control, 6 (1968), 681-697.  doi: 10.1137/0306044.  Google Scholar [30] H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.  doi: 10.3934/jimo.2016016.  Google Scholar [31] W. Zhang, J. Hu and A. Abate, On the value function of the discrete-time switched LQR problem, IEEE Transactions on Automatic Control, 54 (2009), 2669-2674.  doi: 10.1109/TAC.2009.2031574.  Google Scholar [32] W. Zhang, Y. Lin and L. Xue, Linear quadratic Pareto optimal control problem of stochastic singular systems, Journal of the Franklin Institute, 354 (2017), 1220-1238.  doi: 10.1016/j.jfranklin.2016.11.021.  Google Scholar

show all references

##### References:
 [1] K. Avrachenkov, O. Habachi, A. Piunovskiy and Y. Zhang, Infinite horizon optimal impulsive control with applications to Internet congestion control, International Journal of Control, 88 (2015), 703-716.  doi: 10.1080/00207179.2014.971436.  Google Scholar [2] A. V. Balakrishnan, On stochastic bang-bang control, Applied Mathematics and Optimization, 6 (1980), 91-96.  doi: 10.1007/BF01442885.  Google Scholar [3] M. Basin and and J. Rodriguez-Gonzalez, Optimal control for linear systems with time delay in control input, IEEE Transactions on Automatic Control, 51 (2006), 91-97.  doi: 10.1109/TAC.2005.861718.  Google Scholar [4] A. Cairns, Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time, Astin Bulletin, 30 (2000), 19-55.  doi: 10.2143/AST.30.1.504625.  Google Scholar [5] S. L. Campbell, Singular Systems of Differential Equations II, Pitman, Boston, Mass.-London, 1982.  Google Scholar [6] M. Chadli, H. R. Karimi and P. Shi, On stability and stabilization of singular uncertain Takagi-Sugeno fuzzy systems, Journal of the Franklin Institute, 351 (2014), 1453-1463.  doi: 10.1016/j.jfranklin.2013.11.008.  Google Scholar [7] L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica J. IFAC, 46 (2010), 1074-1080.  doi: 10.1016/j.automatica.2010.03.005.  Google Scholar [8] D. Cobb, Controllability, observability, and duality in singular systems, IEEE Transactions on Automatic Control, 29 (1984), 1076-1082.  doi: 10.1109/TAC.1984.1103451.  Google Scholar [9] L. Dai, Singular Control Systems, Springer-Verlag, Berlin, Germany, 1989. doi: 10.1007/BFb0002475.  Google Scholar [10] J.-e Feng, P. Cui and Z. Hou, Singular linear quadratic optimal control for singular stochastic discrete-time systems, Optimal Control Applications and Methods, 34 (2013), 505-516.  doi: 10.1002/oca.2033.  Google Scholar [11] J. Y. Ishihara and M. H. Terra, On the Lyapunov theorem for singular systems, IEEE Transactions on Automatic Control, 47 (2002), 1926-1930.  doi: 10.1109/TAC.2002.804463.  Google Scholar [12] R. E. Kalman, Contribution to the theory of optimal control, Boletin Sociedad Matem$\acute{a}$tica Mexicana, 5 (1960), 102–119.  Google Scholar [13] Y. Kang and Y. Zhu, Bang-bang optimal control for multi-stage uncertain systems, International Journal on Information, 15 (2012), 3229-3237.   Google Scholar [14] D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2012. Google Scholar [15] N. Kumaresan and P. Balasubramaniam, Optimal control for stochastic nonlinear singular system using neural networks, Computers and Mathematics with Applications, 56 (2008), 2145-2154.  doi: 10.1016/j.camwa.2008.03.041.  Google Scholar [16] P. Kunkel and V. Mehrmann, The linear quadratic optimal control problem for linear descriptor systems with variable coefficients, Mathematics of Control, Signals, and Systems, 10 (1997), 247-264.  doi: 10.1007/BF01211506.  Google Scholar [17] B. Li and and Y. Zhu, The piecewise parametric optimal control of uncertain linear quadratic models, International Journal of Systems Science, 50 (2019), 961-969.  doi: 10.1080/00207721.2019.1586003.  Google Scholar [18] F. Liao, Z. Ren, M. Tomizuka and J. Wu, Preview control for impulse-free continuous-time descriptor systems, International Journal of Control, 88 (2015), 1142-1149.  doi: 10.1080/00207179.2014.996769.  Google Scholar [19] J. Lian, C. Li and D. Liu, Input-to-state stability for discrete-time nonlinear switched singular systems, IET Control Theory Appl., 11 (2017), 2893–2899. doi: 10.1049/iet-cta.2017.0028.  Google Scholar [20] W. Liu, X. Liang, Y. Ma and W. Liu, Aircraft trajectory optimization for collision avoidance using stochastic optimal control, Asian Journal of Control, 21 (2019), 2308-2320.  doi: 10.1002/asjc.1855.  Google Scholar [21] D. G. Luenberger, Dynamic equations in descriptor form, IEEE Transactions on Automatic Control, 22 (1977), 312-321.  doi: 10.1109/tac.1977.1101502.  Google Scholar [22] D. G. Luenberger and A. Arbel, Singular dynamic Leontief systems, Econometrica, 45 (1977), 991-995.  doi: 10.2307/1912686.  Google Scholar [23] G. P. Mantas and N. J. Krikelis, LQ optimal control for discrete descriptor systems with a complete set of boundary information, International Journal of Control, 47 (1988), 1467-1477.  doi: 10.1080/00207178808906108.  Google Scholar [24] D. S. Naidu, Singular perturbations and time scales in control theory and applications: An overview., Dynamics of Continuous Discrete & Impulsive Systems Ser. B Appl. Algorithms, 9 (2002), 233-278.   Google Scholar [25] H. Peng, Fei. Li, J. Liu and Z. Ju, A symplectic instantaneous optimal control for robot trajectory tracking with differential-algebraic equation models, IEEE Transactions on Industrial Electronics, 67 (2020), 3819-3829.  doi: 10.1109/TIE.2019.2916390.  Google Scholar [26] Y. Shu and Y. Zhu, Optimistic value based optimal control for uncertain linear singular systems and application to a dynamic input-output model, ISA Transactions, 71 (2017), 235-251.  doi: 10.1016/j.isatra.2017.08.007.  Google Scholar [27] Y. Shu and Y. Zhu, Stability analysis of uncertain singular systems, Soft Computing, 22 (2018), 5671-5681.  doi: 10.1007/s00500-017-2599-2.  Google Scholar [28] K. Wang, L. Xie and Y. Liu, Characteristics of self-excited oscillation in a curved diffuser., Journal of Propulsion Technology, 39 (2018), 1955-1964.   Google Scholar [29] W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM Journal on Control, 6 (1968), 681-697.  doi: 10.1137/0306044.  Google Scholar [30] H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.  doi: 10.3934/jimo.2016016.  Google Scholar [31] W. Zhang, J. Hu and A. Abate, On the value function of the discrete-time switched LQR problem, IEEE Transactions on Automatic Control, 54 (2009), 2669-2674.  doi: 10.1109/TAC.2009.2031574.  Google Scholar [32] W. Zhang, Y. Lin and L. Xue, Linear quadratic Pareto optimal control problem of stochastic singular systems, Journal of the Franklin Institute, 354 (2017), 1220-1238.  doi: 10.1016/j.jfranklin.2016.11.021.  Google Scholar
Trajectories of optimal controls $u^*(k)$ for problem (11)
Trajectories of state vectors $x(k)$ for problem (11)
The optimal results of problem (11)
 Stage $(u^{*}(k))^T$ $\mu_k$ $x_1^T(k)$ $x^T(k)$ $J\big(k, x_1(k), x_2(7)\big)$ 0 $(3.567, -2.323, -5.406)$ $2.517$ $(1, 2)$ $(1.000, -5.434, 2.000, -4.600)$ $-112.722$ 1 $(0.338, -2.541, -0.797)$ $1.743$ $(-4.756, 0.145)$ $(-4.756, -2.699, 0.145, -4.756)$ $-55.708$ 2 $(1.796, -2.599, -0.303)$ $2.098$ $(-1.416, -4.148)$ $(-1.416, -5.019, -4.148, -1.218)$ $20.114$ 3 $(0.167, -1.651, 0.469)$ $0.202$ $(-1.429, -1.614)$ $(-1.429, -1.769, -1.614, 0.490)$ $41.636$ 4 $(0.635, -1.194, 0.582)$ $1.165$ $(-0.931, -2.843)$ $(-0.931, -2.743, -2.843, -0.320)$ $39.171$ 5 $(0.862, -0.338, -0.251)$ $0.872$ $(-0.362, -0.430)$ $(-0.362, -1.988, -0.430, -1.147)$ $32.059$ 6 $(0.390, -0.274, -0.403)$ $1.394$ $(-0.505, 0.182)$ $(-0.505, -2.598, 0.182, -2.027)$ $46.328$ 7 $u^{*}(7)\in R^3$ $(-1.217, 1.619)$ $(-1.217, 0.598, 1.619, -0.217)$ $7.363$
 Stage $(u^{*}(k))^T$ $\mu_k$ $x_1^T(k)$ $x^T(k)$ $J\big(k, x_1(k), x_2(7)\big)$ 0 $(3.567, -2.323, -5.406)$ $2.517$ $(1, 2)$ $(1.000, -5.434, 2.000, -4.600)$ $-112.722$ 1 $(0.338, -2.541, -0.797)$ $1.743$ $(-4.756, 0.145)$ $(-4.756, -2.699, 0.145, -4.756)$ $-55.708$ 2 $(1.796, -2.599, -0.303)$ $2.098$ $(-1.416, -4.148)$ $(-1.416, -5.019, -4.148, -1.218)$ $20.114$ 3 $(0.167, -1.651, 0.469)$ $0.202$ $(-1.429, -1.614)$ $(-1.429, -1.769, -1.614, 0.490)$ $41.636$ 4 $(0.635, -1.194, 0.582)$ $1.165$ $(-0.931, -2.843)$ $(-0.931, -2.743, -2.843, -0.320)$ $39.171$ 5 $(0.862, -0.338, -0.251)$ $0.872$ $(-0.362, -0.430)$ $(-0.362, -1.988, -0.430, -1.147)$ $32.059$ 6 $(0.390, -0.274, -0.403)$ $1.394$ $(-0.505, 0.182)$ $(-0.505, -2.598, 0.182, -2.027)$ $46.328$ 7 $u^{*}(7)\in R^3$ $(-1.217, 1.619)$ $(-1.217, 0.598, 1.619, -0.217)$ $7.363$

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