# American Institute of Mathematical Sciences

September  2018, 8(3&4): 557-582. doi: 10.3934/mcrf.2018023

## Numerical study of polynomial feedback laws for a bilinear control problem

 1 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria 2 RICAM Institute, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria

* Corresponding author: Karl Kunisch

Received  September 2017 Revised  July 2018 Published  September 2018

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

Citation: Tobias Breiten, Karl Kunisch, Laurent Pfeiffer. Numerical study of polynomial feedback laws for a bilinear control problem. Mathematical Control & Related Fields, 2018, 8 (3&4) : 557-582. doi: 10.3934/mcrf.2018023
##### References:
 [1] C. Aguilar and A. Krener, Numerical solutions to the Bellman equation of optimal control, Journal of Optimization Theory and Applications, 160 (2014), 527-552. doi: 10.1007/s10957-013-0403-8. [2] A. Alla and M. Falcone, An adaptive pod approximation method for the control of advectiondiffusion equations, in Control and Optimization with PDE Constraints (eds. K. Bredies, C. Clason, K. Kunisch and G. von Winckel), Springer Basel, Basel, 2013, 1-17. doi: 10.1007/978-3-0348-0631-2_1. [3] F. Ancona and A. Bressan, Flow stability of patchy vector fields and robust feedback stabilization, SIAM Journal on Control and Optimization, 41 (2002), 1455-1476. doi: 10.1137/S0363012901391676. [4] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM Journal on Control and Optimization, 48 (2009), 1797-1830. doi: 10.1137/070682630. [5] P. Benner, T. Breiten, C. Hartmann and B. Schmidt, Model Reduction of Controlled Fokker-Planck and Liouville-von Neumann Equations, Technical report, 2017, Available from https://arXiv.org/abs/1706.09882. [6] P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems, SIAM Journal on Control and Optimization, 49 (2011), 686-711. doi: 10.1137/09075041X. [7] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser Boston Basel Berlin, 2007. doi: 10.1007/978-0-8176-4581-6. [8] T. Breiten, K. Kunisch and L. Pfeiffer, Control strategies for the Fokker-Planck equation, ESAIM: Control, Optimisation and Calculus of Variations, 24 (2018), 741-763. doi: 10.1051/cocv/2017046. [9] T. Breiten, K. Kunisch and L. Pfeiffer, Taylor Expansions for the HJB Equation Associated with a Bilinear Control Problem, Technical report, SFB-Report 2017-006, 2017, Available from http://imsc.uni-graz.at/mobis/publications/SFB-Report-2017-006_v2.pdf. [10] E. Carlini and F.J. Silva, On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications, SIAM J. Numer. Anal., 56 (2018), 2148-2177. doi: 10.1137/17M1143022. [11] J. Chang and G. Cooper, A practical scheme for Fokker--Planck equations, Journal of Computational Physics, 6 (1970), 1-16. [12] T. Damm, Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations, Numerical Linear Algebra with Applications, 15 (2008), 853-871. doi: 10.1002/nla.603. [13] L. Grasedyck, Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure, Computing, 72 (2004), 247-265. doi: 10.1007/s00607-003-0037-z. [14] K. Ito and S. Ravindran, A reduced-order method for simulation and control of fluid flows, Journal of Computational Physics, 143 (1998), 403-425, URL http://www.sciencedirect.com/science/article/pii/S0021999198959435. doi: 10.1006/jcph.1998.5943. [15] 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), A629-A652. doi: 10.1137/17M1116635. [16] A. Krener, C. Aguilar and T. Hunt, Mathematical system theory - festschrift in honor of uwe helmke on the occasion of his sixtieth birthday, CreateSpace, Chapter Series Solutions of HJB Equations, (2013), 247-260. [17] K. Kunisch, S. Volkwein and L. Xie, HJB-POD-based feedback design for the optimal control of evolution problems, SIAM Journal on Applied Dynamical Systems, 3 (2004), 701-722. doi: 10.1137/030600485. [18] B.J. Matkowsky and Z. Schuss, Eigenvalues of the Fokker-Planck operator and the approach to equilibrium for diffusions in potential fields, SIAM Journal on Applied Mathematics, 40 (1981), 242-254. doi: 10.1137/0140020. [19] M. Opmeer, Decay of Hankel singular values of analytic control systems, Systems & Control Letters, 59 (2010), 635-638. doi: 10.1016/j.sysconle.2010.07.009. [20] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier--Stokes equations, SIAM Journal on Control and Optimization, 45 (2006), 790-828. doi: 10.1137/050628726.

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##### References:
 [1] C. Aguilar and A. Krener, Numerical solutions to the Bellman equation of optimal control, Journal of Optimization Theory and Applications, 160 (2014), 527-552. doi: 10.1007/s10957-013-0403-8. [2] A. Alla and M. Falcone, An adaptive pod approximation method for the control of advectiondiffusion equations, in Control and Optimization with PDE Constraints (eds. K. Bredies, C. Clason, K. Kunisch and G. von Winckel), Springer Basel, Basel, 2013, 1-17. doi: 10.1007/978-3-0348-0631-2_1. [3] F. Ancona and A. Bressan, Flow stability of patchy vector fields and robust feedback stabilization, SIAM Journal on Control and Optimization, 41 (2002), 1455-1476. doi: 10.1137/S0363012901391676. [4] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM Journal on Control and Optimization, 48 (2009), 1797-1830. doi: 10.1137/070682630. [5] P. Benner, T. Breiten, C. Hartmann and B. Schmidt, Model Reduction of Controlled Fokker-Planck and Liouville-von Neumann Equations, Technical report, 2017, Available from https://arXiv.org/abs/1706.09882. [6] P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems, SIAM Journal on Control and Optimization, 49 (2011), 686-711. doi: 10.1137/09075041X. [7] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser Boston Basel Berlin, 2007. doi: 10.1007/978-0-8176-4581-6. [8] T. Breiten, K. Kunisch and L. Pfeiffer, Control strategies for the Fokker-Planck equation, ESAIM: Control, Optimisation and Calculus of Variations, 24 (2018), 741-763. doi: 10.1051/cocv/2017046. [9] T. Breiten, K. Kunisch and L. Pfeiffer, Taylor Expansions for the HJB Equation Associated with a Bilinear Control Problem, Technical report, SFB-Report 2017-006, 2017, Available from http://imsc.uni-graz.at/mobis/publications/SFB-Report-2017-006_v2.pdf. [10] E. Carlini and F.J. Silva, On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications, SIAM J. Numer. Anal., 56 (2018), 2148-2177. doi: 10.1137/17M1143022. [11] J. Chang and G. Cooper, A practical scheme for Fokker--Planck equations, Journal of Computational Physics, 6 (1970), 1-16. [12] T. Damm, Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations, Numerical Linear Algebra with Applications, 15 (2008), 853-871. doi: 10.1002/nla.603. [13] L. Grasedyck, Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure, Computing, 72 (2004), 247-265. doi: 10.1007/s00607-003-0037-z. [14] K. Ito and S. Ravindran, A reduced-order method for simulation and control of fluid flows, Journal of Computational Physics, 143 (1998), 403-425, URL http://www.sciencedirect.com/science/article/pii/S0021999198959435. doi: 10.1006/jcph.1998.5943. [15] 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), A629-A652. doi: 10.1137/17M1116635. [16] A. Krener, C. Aguilar and T. Hunt, Mathematical system theory - festschrift in honor of uwe helmke on the occasion of his sixtieth birthday, CreateSpace, Chapter Series Solutions of HJB Equations, (2013), 247-260. [17] K. Kunisch, S. Volkwein and L. Xie, HJB-POD-based feedback design for the optimal control of evolution problems, SIAM Journal on Applied Dynamical Systems, 3 (2004), 701-722. doi: 10.1137/030600485. [18] B.J. Matkowsky and Z. Schuss, Eigenvalues of the Fokker-Planck operator and the approach to equilibrium for diffusions in potential fields, SIAM Journal on Applied Mathematics, 40 (1981), 242-254. doi: 10.1137/0140020. [19] M. Opmeer, Decay of Hankel singular values of analytic control systems, Systems & Control Letters, 59 (2010), 635-638. doi: 10.1016/j.sysconle.2010.07.009. [20] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier--Stokes equations, SIAM Journal on Control and Optimization, 45 (2006), 790-828. doi: 10.1137/050628726.
1D Fokker-Planck equation
Comparison of the original and reduced models, for $n = 100$, $r = 25$, and $\beta = 10^{-4}$
Comparison of the original and reduced models, for $n = 100$, $r = 25$, and $\beta = 10^{-4}$
Comparison of the reduced models with $r = 21$ and $r = 9$, derived from a finer discretization (with $n = 1000$), with the setup of Figure 3
Convergence of the control laws for $\beta = 10^{-4}, n = 1000$ and $r = 9.$
Convergence of the control laws for $\beta = 10^{-4}, n = 1000$ and $r = 9.$
Initial condition and controls for the test case 2
Initial condition and controls for the test case 3
2D Fokker-Planck equation
Control shape functions α1 and α2
Singular value decay for n = 2500
Initial condition and controls for the test case 4
Initial condition and controls for the test case 5
Convergence results for the test case 1
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_6)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.038 0.038 0.038 0.038 0.038 0.038 1e$^{-4}$ 0.034 0.033 0.033 0.033 0.033 0.032 1e$^{-5}$ 0.037 0.031 0.031 0.031 0.031 0.030 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ $p=6$ 1e$^{-3}$ 0.228 0.026 0.024 0.024 0.024 1e$^{-4}$ 4.26 1.19 0.82 0.61 0.61 1e$^{-5}$ 29.8 10.3 7.91 4.70 4.05 (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_6)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.038 0.038 0.038 0.038 0.038 0.038 1e$^{-4}$ 0.034 0.033 0.033 0.033 0.033 0.032 1e$^{-5}$ 0.037 0.031 0.031 0.031 0.031 0.030 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ $p=6$ 1e$^{-3}$ 0.228 0.026 0.024 0.024 0.024 1e$^{-4}$ 4.26 1.19 0.82 0.61 0.61 1e$^{-5}$ 29.8 10.3 7.91 4.70 4.05 (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
Convergence results for the test case 2
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_6)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.156 0.155 0.155 0.155 0.155 0.154 5e$^{-4}$ 0.147 0.145 0.145 0.145 0.145 0.144 1e$^{-4}$ 0.138 0.122 0.120 0.120 0.120 0.119 5e$^{-5}$ 0.190 0.114 0.111 0.112 0.111 0.110 1e$^{-5}$ 0.205 0.194 0.104 0.111 0.113 0.095 (A) Cost of the controls up. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ $p=6$ 1e$^{-3}$ 1.149 0.169 0.119 0.034 0.031 5e$^{-4}$ 2.583 0.737 0.171 0.336 0.219 1e$^{-4}$ 18.50 7.02 3.16 4.01 1.52 5e$^{-5}$ 46.87 13.18 8.40 8.17 2.65 1e$^{-5}$ 90.5 78.0 39.0 42.6 34.3 (B) L2-distance between the controls up and the optimal control uopt.
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_6)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.156 0.155 0.155 0.155 0.155 0.154 5e$^{-4}$ 0.147 0.145 0.145 0.145 0.145 0.144 1e$^{-4}$ 0.138 0.122 0.120 0.120 0.120 0.119 5e$^{-5}$ 0.190 0.114 0.111 0.112 0.111 0.110 1e$^{-5}$ 0.205 0.194 0.104 0.111 0.113 0.095 (A) Cost of the controls up. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ $p=6$ 1e$^{-3}$ 1.149 0.169 0.119 0.034 0.031 5e$^{-4}$ 2.583 0.737 0.171 0.336 0.219 1e$^{-4}$ 18.50 7.02 3.16 4.01 1.52 5e$^{-5}$ 46.87 13.18 8.40 8.17 2.65 1e$^{-5}$ 90.5 78.0 39.0 42.6 34.3 (B) L2-distance between the controls up and the optimal control uopt.
Convergence results for the test case 3
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_6)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.525 0.511 0.511 0.512 0.510 0.507 5e$^{-4}$ 0.451 0.417 0.431 0.459 0.446 0.408 1e$^{-4}$ 0.381 0.368 2.689 $\infty$ $\infty$ 0.246 5e$^{-5}$ 0.381 0.432 $\infty$ $\infty$ $\infty$ 0.206 1e$^{-5}$ 0.365 $\infty$ $\infty$ $\infty$ $\infty$ 0.147 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ $p=6$ 1e$^{-3}$ 4.88 1.50 1.77 2.31 1.52 5e$^{-4}$ 11.26 5.03 7.11 11.89 11.99 1e$^{-4}$ 46.34 35.36 57.08 $\infty$ $\infty$ 5e$^{-5}$ 74.79 60.86 $\infty$ $\infty$ $\infty$ 1e$^{-5}$ 172.3 $\infty$ $\infty$ $\infty$ $\infty$ (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_6)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.525 0.511 0.511 0.512 0.510 0.507 5e$^{-4}$ 0.451 0.417 0.431 0.459 0.446 0.408 1e$^{-4}$ 0.381 0.368 2.689 $\infty$ $\infty$ 0.246 5e$^{-5}$ 0.381 0.432 $\infty$ $\infty$ $\infty$ 0.206 1e$^{-5}$ 0.365 $\infty$ $\infty$ $\infty$ $\infty$ 0.147 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ $p=6$ 1e$^{-3}$ 4.88 1.50 1.77 2.31 1.52 5e$^{-4}$ 11.26 5.03 7.11 11.89 11.99 1e$^{-4}$ 46.34 35.36 57.08 $\infty$ $\infty$ 5e$^{-5}$ 74.79 60.86 $\infty$ $\infty$ $\infty$ 1e$^{-5}$ 172.3 $\infty$ $\infty$ $\infty$ $\infty$ (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
Convergence results for the test case 4
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.247 0.235 0.234 0.234 0.232 5e$^{-4}$ 0.232 0.207 0.205 0.205 0.203 1e$^{-4}$ 0.252 0.180 0.174 0.174 0.171 5e$^{-5}$ 0.279 0.179 0.168 0.168 0.165 1e$^{-5}$ 0.524 0.182 20.696 0.164 0.158 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ 1e$^{-3}$ 3.53 0.80 0.19 0.14 5e$^{-4}$ 6.73 1.42 0.37 0.24 1e$^{-4}$ 27.40 5.78 1.83 1.24 5e$^{-5}$ 52.50 11.06 3.69 2.40 1e$^{-5}$ 257.01 63.97 84.31 10.61 (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_{\text{opt}})$ 1e$^{-3}$ 0.247 0.235 0.234 0.234 0.232 5e$^{-4}$ 0.232 0.207 0.205 0.205 0.203 1e$^{-4}$ 0.252 0.180 0.174 0.174 0.171 5e$^{-5}$ 0.279 0.179 0.168 0.168 0.165 1e$^{-5}$ 0.524 0.182 20.696 0.164 0.158 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ 1e$^{-3}$ 3.53 0.80 0.19 0.14 5e$^{-4}$ 6.73 1.42 0.37 0.24 1e$^{-4}$ 27.40 5.78 1.83 1.24 5e$^{-5}$ 52.50 11.06 3.69 2.40 1e$^{-5}$ 257.01 63.97 84.31 10.61 (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
Convergence results for the test case 5
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_{\text{opt}})$ 1e$^{-1}$ 7.58 7.57 7.57 7.57 7.52 5e$^{-2}$ 6.41 6.39 6.40 6.39 6.35 1e$^{-2}$ 3.70 3.34 3.09 3.32 3.00 5e$^{-3}$ 3.07 2.68 2.28 2.96 2.05 1e$^{-3}$ 2.45 2.41 $\infty$ $\infty$ 0.93 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ 1e$^{-1}$ 0.70 0.61 0.62 0.60 5e$^{-2}$ 1.10 0.69 0.80 0.63 1e$^{-2}$ 13.02 11.10 4.08 9.01 5e$^{-3}$ 21.59 19.80 9.66 20.06 1e$^{-3}$ 47.34 55.69 $\infty$ $\infty$ (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
 $\beta$ $J(u_2)$ $J(u_3)$ $J(u_4)$ $J(u_5)$ $J(u_{\text{opt}})$ 1e$^{-1}$ 7.58 7.57 7.57 7.57 7.52 5e$^{-2}$ 6.41 6.39 6.40 6.39 6.35 1e$^{-2}$ 3.70 3.34 3.09 3.32 3.00 5e$^{-3}$ 3.07 2.68 2.28 2.96 2.05 1e$^{-3}$ 2.45 2.41 $\infty$ $\infty$ 0.93 (A) Cost of the controls $u_p$. $\beta$ $\| u_p-u_{\text{opt}} \|_{L^2(0, T)}$ $p=2$ $p=3$ $p=4$ $p=5$ 1e$^{-1}$ 0.70 0.61 0.62 0.60 5e$^{-2}$ 1.10 0.69 0.80 0.63 1e$^{-2}$ 13.02 11.10 4.08 9.01 5e$^{-3}$ 21.59 19.80 9.66 20.06 1e$^{-3}$ 47.34 55.69 $\infty$ $\infty$ (B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.
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