doi: 10.3934/era.2021046
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A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

2. 

Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

* Corresponding author: Bing Sun

Received  November 2020 Revised  May 2021 Early access June 2021

Fund Project: The second author is supported in part by the National Natural Science Foundation of China under Grant No. 11471036

In this paper, we present a feedback design for numerical solution to optimal control problems, which is based on solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation. An upwind finite-difference scheme is adopted to solve the HJB equation under the framework of the dynamic programming viscosity solution (DPVS) approach. Different from the usual existing algorithms, the numerical control function is interpolated in turn to gain the approximation of optimal feedback control-trajectory pair. Five simulations are executed and both of them, without exception, output the accurate numerical results. The design can avoid solving the HJB equation repeatedly, thus efficaciously promote the computation efficiency and save memory.

Citation: Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, doi: 10.3934/era.2021046
References:
[1]

A. Alla, Model Reduction for A Dynamic Programming Approach to Optimal Control Problems with PDE Constraints, Ph.D thesis, University of Rome in Sapienza, Italy, 2014. Google Scholar

[2]

A. Alla, B. Haasdonk and A. Schmidt, Feedback control of parametrized PDEs via model order reduction and dynamic programming principle, Adv. Comput. Math., 46 (2020), Paper No. 9, 28 pp. doi: 10.1007/s10444-020-09744-8.  Google Scholar

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, with appendices by Maurizio Falcone and Pierpaolo Soravia, Birkhäuser, Boston, 1997., doi: 10.1007/978-0-8176-4755-1.  Google Scholar

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M. V. Basin and M. A. Pinsky, Control of Kalman-like filters using impulse and continuous feedback design, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 169-184.  doi: 10.3934/dcdsb.2002.2.169.  Google Scholar

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T. Breiten and K. Kunisch, Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations, Discrete Contin. Dyn. Syst., 40 (2020), 4197-4229.  doi: 10.3934/dcds.2020178.  Google Scholar

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D. CastorinaA. Cesaroni and L. Rossi, On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary, Commun. Pure Appl. Anal., 15 (2016), 1251-1263.  doi: 10.3934/cpaa.2016.15.1251.  Google Scholar

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M. G. CrandallL. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502.  doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

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[14]

B.-Z. Guo and B. Sun, Dynamic programming approach to the numerical solution of optimal control with paradigm by a mathematical model for drug therapies of HIV/AIDS, Optim. Eng., 15 (2004), 119-136.  doi: 10.1007/s11081-012-9204-4.  Google Scholar

[15]

B.-Z. Guo and B. Sun, Numerical solution to the optimal feedback control of continuous casting process, J. Global Optim., 39 (2007), 171-195.  doi: 10.1007/s10898-006-9130-0.  Google Scholar

[16]

Y.-C. Ho, On centralized optimal control, IEEE Trans. Automat. Control, 50 (2005), 537-538.  doi: 10.1109/TAC.2005.844898.  Google Scholar

[17]

D. Kalise and K. Kunisch, Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs, SIAM J. Sci. Comput., 40 (2018), 629-652.  doi: 10.1137/17M1116635.  Google Scholar

[18]

K. KunischS. Volkwein and L. Xie, HJB-POD-based feedback design for the optimal control of evolution problems, SIAM J. Appl. Dyn. Syst., 3 (2004), 701-722.  doi: 10.1137/030600485.  Google Scholar

[19]

K. Kunisch and L. Xie, POD-based feedback control of the Burgers equation to solving the evolutionary HJB equation, Comput. Math. Appl., 49 (2005), 1113-1126.  doi: 10.1016/j.camwa.2004.07.022.  Google Scholar

[20]

K. Kunisch and L. Xie, Suboptimal feedback control of flow separation by POD model reduction, in Real-Time PDE-Constrained Optimization, (eds. L. T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes and B. van Bloemen Waanders), Computational Science & Engineering, SIAM, (2007), 233–250. doi: 10.1137/1.9780898718935.ch12.  Google Scholar

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H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[22]

W. M. McEneaney, Convergence rate for a curse-of-dimensionality-free method for Hamilton-Jacobi-Bellman PDEs represented as maxima of quadratic forms, SIAM J. Control Optim., 48 (2009), 2651-2685.  doi: 10.1137/070687980.  Google Scholar

[23]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York, 1962.  Google Scholar

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T. Sauer, Numerical Analysis, 2$^nd$ edition, Pearson Education, Essex, 2012. Google Scholar

[26]

C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.  doi: 10.3934/dcds.2015.35.4639.  Google Scholar

[27]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Texts in Applied Mathematics, Vol. 12, Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[28]

B. Sun and B.-Z. Guo, Convergence of an upwind finite-difference scheme for Hamilton-Jacobi-Bellman equation in optimal control, IEEE Trans. Automat. Control, 60 (2015), 3012-3017.  doi: 10.1109/TAC.2015.2406976.  Google Scholar

[29]

S. WangF. Gao and K. L. Teo, An upwind finite-difference method for the approximation of viscosity solutions to Hamilton-Jacobi-Bellman equations, IMA J. Math. Control Inform., 17 (2000), 167-178.  doi: 10.1093/imamci/17.2.167.  Google Scholar

[30] J. M. Yong and X. J. Li, A Concise Lecture Note on Optimal Control Theory, Higher Education Press, Beijing, 2006.   Google Scholar
[31]

X. Y. Zhou, Verification theorems within the framework of viscosity solutions, J. Math. Anal. Appl., 177 (1993), 208-225.  doi: 10.1006/jmaa.1993.1253.  Google Scholar

show all references

References:
[1]

A. Alla, Model Reduction for A Dynamic Programming Approach to Optimal Control Problems with PDE Constraints, Ph.D thesis, University of Rome in Sapienza, Italy, 2014. Google Scholar

[2]

A. Alla, B. Haasdonk and A. Schmidt, Feedback control of parametrized PDEs via model order reduction and dynamic programming principle, Adv. Comput. Math., 46 (2020), Paper No. 9, 28 pp. doi: 10.1007/s10444-020-09744-8.  Google Scholar

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, with appendices by Maurizio Falcone and Pierpaolo Soravia, Birkhäuser, Boston, 1997., doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[4]

M. V. Basin and M. A. Pinsky, Control of Kalman-like filters using impulse and continuous feedback design, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 169-184.  doi: 10.3934/dcdsb.2002.2.169.  Google Scholar

[5]

T. Breiten and K. Kunisch, Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations, Discrete Contin. Dyn. Syst., 40 (2020), 4197-4229.  doi: 10.3934/dcds.2020178.  Google Scholar

[6]

D. CastorinaA. Cesaroni and L. Rossi, On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary, Commun. Pure Appl. Anal., 15 (2016), 1251-1263.  doi: 10.3934/cpaa.2016.15.1251.  Google Scholar

[7]

M. G. Crandall, Viscosity solutions: A primer, in Lecture Notes in Mathematics (eds. I. Capuzzo-Dolcetta and P. L. Lions), Springer-Verlag, Berlin, (1997), 1–43. Google Scholar

[8]

M. G. CrandallL. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502.  doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[9]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[10]

G. Fabrini, M. Falcone and S. Volkwein, Coupling MPC and HJB for the computation of POD-based Feedback Laws, Numerical Mathematics and Advanced Applications—ENUMATH, (2017), 941–949, Lect. Notes Comput. Sci. Eng., 126, Springer, Cham, 2019.  Google Scholar

[11]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, Philadelphia, 2014.  Google Scholar

[12]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2006.  Google Scholar

[13]

S. Gombao, Approximation of optimal controls for semilinear parabolic PDE by solving Hamilton-Jacobi-Bellman equations, in Electronic Proceedings of Fifteenth International Symposium on Mathematical Theory of Networks and Systems (eds. D. S. Gilliam and J. Rosenthal), South Bend, USA, (2002), 1–15. Google Scholar

[14]

B.-Z. Guo and B. Sun, Dynamic programming approach to the numerical solution of optimal control with paradigm by a mathematical model for drug therapies of HIV/AIDS, Optim. Eng., 15 (2004), 119-136.  doi: 10.1007/s11081-012-9204-4.  Google Scholar

[15]

B.-Z. Guo and B. Sun, Numerical solution to the optimal feedback control of continuous casting process, J. Global Optim., 39 (2007), 171-195.  doi: 10.1007/s10898-006-9130-0.  Google Scholar

[16]

Y.-C. Ho, On centralized optimal control, IEEE Trans. Automat. Control, 50 (2005), 537-538.  doi: 10.1109/TAC.2005.844898.  Google Scholar

[17]

D. Kalise and K. Kunisch, Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs, SIAM J. Sci. Comput., 40 (2018), 629-652.  doi: 10.1137/17M1116635.  Google Scholar

[18]

K. KunischS. Volkwein and L. Xie, HJB-POD-based feedback design for the optimal control of evolution problems, SIAM J. Appl. Dyn. Syst., 3 (2004), 701-722.  doi: 10.1137/030600485.  Google Scholar

[19]

K. Kunisch and L. Xie, POD-based feedback control of the Burgers equation to solving the evolutionary HJB equation, Comput. Math. Appl., 49 (2005), 1113-1126.  doi: 10.1016/j.camwa.2004.07.022.  Google Scholar

[20]

K. Kunisch and L. Xie, Suboptimal feedback control of flow separation by POD model reduction, in Real-Time PDE-Constrained Optimization, (eds. L. T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes and B. van Bloemen Waanders), Computational Science & Engineering, SIAM, (2007), 233–250. doi: 10.1137/1.9780898718935.ch12.  Google Scholar

[21]

H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[22]

W. M. McEneaney, Convergence rate for a curse-of-dimensionality-free method for Hamilton-Jacobi-Bellman PDEs represented as maxima of quadratic forms, SIAM J. Control Optim., 48 (2009), 2651-2685.  doi: 10.1137/070687980.  Google Scholar

[23]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York, 1962.  Google Scholar

[24] J. E. Rubio, Control and Optimization: The Linear Treatment of Nonlinear Problems (Nonlinear Science: Theory and Applications), Manchester University Press, Manchester, 1986.   Google Scholar
[25]

T. Sauer, Numerical Analysis, 2$^nd$ edition, Pearson Education, Essex, 2012. Google Scholar

[26]

C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.  doi: 10.3934/dcds.2015.35.4639.  Google Scholar

[27]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Texts in Applied Mathematics, Vol. 12, Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[28]

B. Sun and B.-Z. Guo, Convergence of an upwind finite-difference scheme for Hamilton-Jacobi-Bellman equation in optimal control, IEEE Trans. Automat. Control, 60 (2015), 3012-3017.  doi: 10.1109/TAC.2015.2406976.  Google Scholar

[29]

S. WangF. Gao and K. L. Teo, An upwind finite-difference method for the approximation of viscosity solutions to Hamilton-Jacobi-Bellman equations, IMA J. Math. Control Inform., 17 (2000), 167-178.  doi: 10.1093/imamci/17.2.167.  Google Scholar

[30] J. M. Yong and X. J. Li, A Concise Lecture Note on Optimal Control Theory, Higher Education Press, Beijing, 2006.   Google Scholar
[31]

X. Y. Zhou, Verification theorems within the framework of viscosity solutions, J. Math. Anal. Appl., 177 (1993), 208-225.  doi: 10.1006/jmaa.1993.1253.  Google Scholar

Figure 1.  Numerical and exact solutions in Example 1 with initial state $ y_0 = - \frac{1}{2} $
Figure 2.  Numerical and exact solutions in Example 1 with initial state $ y_0 = \frac{1}{2} $
Figure 3.  Numerical and exact solutions in Example 2 with initial state $ y_0 = 1 $
Figure 4.  Numerical and exact solutions in Example 3 with initial states $ (x_0,y_0) = \left(- \frac{1}{2}, - \frac{1}{2}\right) $
Figure 5.  Numerical and exact solutions in Example 3 with initial states $ (x_0,y_0) = \left(\frac{1}{2}, \frac{1}{2}\right) $
Figure 6.  Numerical solutions of HJB equation (3.6) in Example 4.
Figure 7.  Numerical solutions of HJB equation (3.6) in Example 4 at $ x_1 = 0 $
Figure 8.  Numerical solutions of the HJB equation (3.6) in Example 4 at $ t = \frac{1}{2} $
Figure 9.  Numerical and exact solutions in Example 4 with initial states $ (x_{10},x_{20}) = \left(\frac{1}{2}, - \frac{1}{2}\right) $ (Linear interpolation case)
Figure 10.  Numerical and exact solutions in Example 4 with initial states $ (x_{10},x_{20}) = \left(\frac{1}{2}, - \frac{1}{2}\right) $ (Cubic spline interpolation case)
Figure 11.  Numerical and exact solutions in Example 5 (Linear interpolation case)
Table 1.  The maximum norm error estimates and CPU times for Example 1 with initial state $ y_0 = -\frac{1}{2} $ and $ y_0 = \frac{1}{2} $, respectively
Algorithm Initial state Control Trajectory CPU(s)
Interpolation $ y_0 = - \frac{1}{2} $ 6.6000e-05 3.3001e-05 0.005615
$ y_0 = \frac{1}{2} $ 6.6000e-05 0.016695 0.005484
Algorithm 1 $ y_0 = -\frac{1}{2} $ 6.6000e-05 3.3001e-05 0.015370
$ y_0 = \frac{1}{2} $ 6.6000e-05 0.016695 0.016507
Algorithm 2 $ y_0 = -\frac{1}{2} $ 6.6000e-05 3.3001e-05 0.025182
$ y_0 = \frac{1}{2} $ 6.6000e-05 0.016695 0.025733
Algorithm Initial state Control Trajectory CPU(s)
Interpolation $ y_0 = - \frac{1}{2} $ 6.6000e-05 3.3001e-05 0.005615
$ y_0 = \frac{1}{2} $ 6.6000e-05 0.016695 0.005484
Algorithm 1 $ y_0 = -\frac{1}{2} $ 6.6000e-05 3.3001e-05 0.015370
$ y_0 = \frac{1}{2} $ 6.6000e-05 0.016695 0.016507
Algorithm 2 $ y_0 = -\frac{1}{2} $ 6.6000e-05 3.3001e-05 0.025182
$ y_0 = \frac{1}{2} $ 6.6000e-05 0.016695 0.025733
Table 2.  The maximum norm error estimates for Example 2 with initial state $ y_0 = 1 $
Error Control Trajectory
$ y_0=1 $ 0.025896 0.017297
Error Control Trajectory
$ y_0=1 $ 0.025896 0.017297
Table 3.  The maximum norm error estimates for Example 3 with initial states $ (x_0,y_0) = \left(- \frac{1}{2}, - \frac{1}{2}\right) $ and $ (x_0, y_0) = \left(\frac{1}{2}, \frac{1}{2}\right) $, respectively
Errors Control Trajectory
$ (x_0,y_0) = \left(- \frac{1}{2}, - \frac{1}{2}\right) $ 6.6000e-05 4.6671e-05
$ (x_0,y_0) = \left(\frac{1}{2}, \frac{1}{2}\right) $ 6.6000e-05 0.023611
Errors Control Trajectory
$ (x_0,y_0) = \left(- \frac{1}{2}, - \frac{1}{2}\right) $ 6.6000e-05 4.6671e-05
$ (x_0,y_0) = \left(\frac{1}{2}, \frac{1}{2}\right) $ 6.6000e-05 0.023611
Table 4.  The maximum norm error estimates for Example 4 with initial states $ (x_{10},x_{20}) = \left(\frac{1}{2}, - \frac{1}{2}\right) $ in four interpolation cases
Error Control Trajectory
Linear 0.038706 0.02681
Cubic spline 0.038706 0.026801
Nearest-neighbor 0.069388 0.030588
Cubic-Hermite 0.038706 0.026804
Error Control Trajectory
Linear 0.038706 0.02681
Cubic spline 0.038706 0.026801
Nearest-neighbor 0.069388 0.030588
Cubic-Hermite 0.038706 0.026804
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