doi: 10.3934/mcrf.2021034
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Inverse optimal control of regime-switching jump diffusions

1. 

School of Mathematics, Southeast University, Nanjing 211189, China

2. 

Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea

3. 

School of Mathematics & Statistics, Anhui Normal University, Wuhu 241000, China

* Corresponding author: Jinde Cao

Received  November 2020 Revised  February 2021 Early access June 2021

Fund Project: This work was jointly supported by the Key Project of National Science Foundation of China under Grant No. 61833005 and the National Natural Science Foundation of China under Grant No. 11871076

This paper studies the inverse optimal control using Legendre-Fenchel (in short, LF) translation method for regime-switching jump diffusions. Our approach is to first design inverse pre-optimal stabilization controllers and then obtain inverse optimal stabilizers, which avoids solving a Hamilton-Jacobi-Bellman equation. Finally, an application to stochastic Hamiltonian systems with Markov regime-switching is studied in detail for illustration.

Citation: Wensheng Yin, Jinde Cao, Yong Ren. Inverse optimal control of regime-switching jump diffusions. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021034
References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692.

[3]

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375.  doi: 10.1016/j.jmaa.2012.02.043.

[4]

J. Bao and J. Shao, Permance and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512.

[5]

Z. ChaoK. WangC. Zhu and Y. Zhu, Almost sure and moment exponential stability of regime-switching jump diffusions, SIAM J. Control Optim., 55 (2017), 3458-3488.  doi: 10.1137/16M1082470.

[6]

H. Deng and M. Krstic, Stochastic nonlinear stabilization Ⅰ: A backstepping design, Systems Control Lett., 32 (1997), 143-150.  doi: 10.1016/S0167-6911(97)00068-6.

[7]

H. Deng and M. Krstić, Stochastic nonlinear stabilization-Ⅱ: Inverse optimality, Systems Control Lett., 32 (1997), 151-159.  doi: 10.1016/S0167-6911(97)00067-4.

[8]

K. D. Do, Global inverse optimal stabilization of stochastic nonholonomic systems, Systems Control Lett., 75 (2015), 41-55.  doi: 10.1016/j.sysconle.2014.11.003.

[9]

K. D. Do, Inverse optimal control of stochastic systems driven by Lévy processes, Automatica, 107 (2019), 539-550.  doi: 10.1016/j.automatica.2019.06.016.

[10]

R. A. Freeman and P. V. Kokotovic, Inverse optimality in robust stabilization, SIAM J. Control Optim., 34 (1996), 1365-1391.  doi: 10.1137/S0363012993258732.

[11] G. HardyJ. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1989. 
[12]

L. Hu and X. Mao, Almost sure exponential stabilisation of stochastic systems by state-feedback control, Automatica J. IFAC, 44 (2008), 465-471.  doi: 10.1016/j.automatica.2007.05.027.

[13]

J. HuW. LiuF. Deng and X. Mao, Advances in stabilization of hybrid stochastic differential equations by delay feedback control, SIAM J. Control Optim., 58 (2020), 735-754.  doi: 10.1137/19M1270240.

[14]

R. C. Merton, Continuous-time finance, The Journal of Finance, 46 (1991), 1567-1570.  doi: 10.2307/2328875.

[15]

X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.

[16]

B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Second edition., Universitext, Springer-Verlag, Berlin, 2005.

[17]

J. Shao, Stabilization of regime-switching processes by feedback control based on discrete time observations, SIAM J. Control Optim., 55 (2017), 724-740.  doi: 10.1137/16M1066336.

[18]

J. Shao and F. Xi, Stabilization of regime-switching processes by feedback control based on discrete time observations Ⅱ: State-dependent case, SIAM J. Control Optim., 57 (2019), 1413-1439.  doi: 10.1137/18M1202992.

[19]

H. Ji, J. Shao and F. Xi, Stability of regime-switching jump diffusion processes, J. Math. Anal. Appl., 484 (2020), 21pp. doi: 10.1137/18M1202992.

[20]

F. WuG. Yin and Z. Jin, Kolmogorov-type systems with regime-switching jump diffusion perturbations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2293-2319.  doi: 10.3934/dcdsb.2016048.

[21]

F. Xi, On the stability of jump-diffusions with Markovian switching, J. Math. Anal. Appl., 341 (2008), 588-600.  doi: 10.1016/j.jmaa.2007.10.018.

[22]

F. Xi and C. Zhu, On Feller and strong feller properties and exponential ergodicity of regime switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818.  doi: 10.1137/16M1087837.

[23]

G. Yin and F. Xi, Stability of regime-switching jump diffusions, SIAM J. Control Optim., 48 (2010), 4525-4549.  doi: 10.1137/080738301.

[24]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

[25]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692.

[3]

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375.  doi: 10.1016/j.jmaa.2012.02.043.

[4]

J. Bao and J. Shao, Permance and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512.

[5]

Z. ChaoK. WangC. Zhu and Y. Zhu, Almost sure and moment exponential stability of regime-switching jump diffusions, SIAM J. Control Optim., 55 (2017), 3458-3488.  doi: 10.1137/16M1082470.

[6]

H. Deng and M. Krstic, Stochastic nonlinear stabilization Ⅰ: A backstepping design, Systems Control Lett., 32 (1997), 143-150.  doi: 10.1016/S0167-6911(97)00068-6.

[7]

H. Deng and M. Krstić, Stochastic nonlinear stabilization-Ⅱ: Inverse optimality, Systems Control Lett., 32 (1997), 151-159.  doi: 10.1016/S0167-6911(97)00067-4.

[8]

K. D. Do, Global inverse optimal stabilization of stochastic nonholonomic systems, Systems Control Lett., 75 (2015), 41-55.  doi: 10.1016/j.sysconle.2014.11.003.

[9]

K. D. Do, Inverse optimal control of stochastic systems driven by Lévy processes, Automatica, 107 (2019), 539-550.  doi: 10.1016/j.automatica.2019.06.016.

[10]

R. A. Freeman and P. V. Kokotovic, Inverse optimality in robust stabilization, SIAM J. Control Optim., 34 (1996), 1365-1391.  doi: 10.1137/S0363012993258732.

[11] G. HardyJ. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1989. 
[12]

L. Hu and X. Mao, Almost sure exponential stabilisation of stochastic systems by state-feedback control, Automatica J. IFAC, 44 (2008), 465-471.  doi: 10.1016/j.automatica.2007.05.027.

[13]

J. HuW. LiuF. Deng and X. Mao, Advances in stabilization of hybrid stochastic differential equations by delay feedback control, SIAM J. Control Optim., 58 (2020), 735-754.  doi: 10.1137/19M1270240.

[14]

R. C. Merton, Continuous-time finance, The Journal of Finance, 46 (1991), 1567-1570.  doi: 10.2307/2328875.

[15]

X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.

[16]

B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Second edition., Universitext, Springer-Verlag, Berlin, 2005.

[17]

J. Shao, Stabilization of regime-switching processes by feedback control based on discrete time observations, SIAM J. Control Optim., 55 (2017), 724-740.  doi: 10.1137/16M1066336.

[18]

J. Shao and F. Xi, Stabilization of regime-switching processes by feedback control based on discrete time observations Ⅱ: State-dependent case, SIAM J. Control Optim., 57 (2019), 1413-1439.  doi: 10.1137/18M1202992.

[19]

H. Ji, J. Shao and F. Xi, Stability of regime-switching jump diffusion processes, J. Math. Anal. Appl., 484 (2020), 21pp. doi: 10.1137/18M1202992.

[20]

F. WuG. Yin and Z. Jin, Kolmogorov-type systems with regime-switching jump diffusion perturbations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2293-2319.  doi: 10.3934/dcdsb.2016048.

[21]

F. Xi, On the stability of jump-diffusions with Markovian switching, J. Math. Anal. Appl., 341 (2008), 588-600.  doi: 10.1016/j.jmaa.2007.10.018.

[22]

F. Xi and C. Zhu, On Feller and strong feller properties and exponential ergodicity of regime switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818.  doi: 10.1137/16M1087837.

[23]

G. Yin and F. Xi, Stability of regime-switching jump diffusions, SIAM J. Control Optim., 48 (2010), 4525-4549.  doi: 10.1137/080738301.

[24]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

[25]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.

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