# American Institute of Mathematical Sciences

March 2018, 8(1): 17-46. doi: 10.3934/naco.2018002

## Fourier-splitting method for solving hyperbolic LQR problems

 1 Institute of Mathematics, Eötvös Loránd University Budapest, MTA-ELTE Numerical Analysis and Large Networks Research Group, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary 2 School of Mathematical Sciences and Information Technology, Yachay Tech, Hacienda San José y Proyecto Yachay, EC100650 Urcuquí, Ecuador 3 Department of Mathematics, University of Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria

* Corresponding author: P. Csomós

Received  December 2016 Revised  November 2017 Published  March 2018

Fund Project: The first author is supported by the National Research, Development and Innovation Fund (Hungary) under the grant PD121117

We consider the numerical approximation to linear quadratic regulator problems for hyperbolic partial differential equations where the dynamics is driven by a strongly continuous semigroup. The optimal control is given in feedback form in terms of Riccati operator equations. The computational cost relies on solving the associated Riccati equation and computing the optimal state. In this paper we propose a novel approach based on operator splitting idea combined with Fourier's method to efficiently compute the optimal state. The Fourier's method allows to accurately approximate the exact flow making our approach computational efficient. Numerical experiments in one and two dimensions show the performance of the proposed method.

Citation: Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
##### References:
 [1] H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser, Basel, Switzerland, 2003. [2] A. H. Al-Mohy and N. J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput., 33 (2011), 488-511. [3] E. Arias, V. Hernández, J. Ibanes and J. Peinado, A family of BDF algorithms for solving differential matrix Riccati equations using adaptive techniques, Procedia Computer Science, 1 (2010), 2569-2577. [4] E. Armstrong, An extension of Bass' algorithm for stabilizing linear continuous constant systems, IEEE Trans. Automatic Control, AC-20 (1975), 153-154. [5] A. Balakrishnan, Applied Functional Analysis, Springer-Verlag, New York, 1981. [6] H. Banks, R. Smith and Y. Wang, The modeling of piezoceramic patch interactions with shells, plates and beams, Quart. Appl. Math., 53 (1995), 353-381. [7] A. Bátkai, P. Csomós, B. Farkas and G. Nickel, Operator splitting for non-autonomous evolution equations, J. Funct. Anal., 260 (2011), 2163-2192. [8] A. Bátkai, P. Csomós and G. Nickel, Operator splittings and spatial approximations for evolution equations, J. Evol. Eqs., 9 (2009), 613-636. [9] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, 1993. [10] P. Benner, P. Ezzatti, H. Mena, E. S. Quintana-Ortí and A. Remón, Solving matrix equations on multi-core and many-core architectures, Algorithms, 6 (2013), 857-870. [11] P. Benner and H. Mena, Numerical solution of the infinite-dimensional LQR-problem and the associated differential Riccati equations, Journal of Numerical Mathematics (2016), in press. [12] P. Benner and H. Mena, Rosenbrock methods for solving differential Riccati equations, IEEE Transactions on Automatic Control, 58 (2013), 2950-2957. [13] P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM Mitteilungen, 36 (2013), 32-52. [14] P. Csomós and J. Winckler, A semigroup proof for the well-posedness of the linearised shallow water equations, J. Anal. Math., 43 (2017), 445-459. [15] G. Da Prato, Direct solution of a Riccati equation arising in stochastic control theory, Appl. Math. Optim., 11 (1984), 191-208. [16] G. Da Prato, P. Kunstmann, I. Lasiecka, A. Lunardi, R. Schnaubelt and L. Weis, Functional Analytic Methods for Evolution Equations, Springer-Verlag, Berlin, 2004. [17] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. [18] F. Flandoli, Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary, Appl. Math. Optim., 14 (1986), 107-129. [19] C. Hafizoglu, I. Lasiecka, T. Levajković, H. Mena and A. Tuffaha, The stochastic linear quadratic problem with singular estimates, SIAM J. Control Optim., 55 (2017), 595-626. [20] E. Hansen and A. Ostermann, Exponential splitting for unbounded operators, Math. Comput., 78 (2009), 1485-1496. [21] A. Ichikawa, Dynamic programming approach to stochastic evolution equation, SIAM J. Control. Optim., 17 (1979), 152-174. [22] A. Ichikawa and H. Katayama, Remarks on the time-varying H∞ Riccati equations, Sys. Cont. Lett., 37 (1999), 335-345. [23] O. Iftime and M. Opmeer, A representation of all bounded selfadjoint solutions of the algebraic Riccati equation for systems with an unbounded observation operator, Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 14-07 (2004), 2865-2870. [24] K. Ito and F. Kappel, Evolution Equations and Approximations, World Scientific, Singapore, 2002. [25] T. Jahnke and Ch. Lubich, Error bounds for exponential operator splittings, BIT, 40 (2000), 735-744. [26] D. Kleinman, On an iterative technique for Riccati equation computations, IEEE Trans. Automatic Control, AC-13 (1968), 114-115. [27] A. Kofler, H. Mena and A. Ostermann, Splitting methods for stochastic partial differential equations, preprint [28] N. Lang, H. Mena and J. Saak, On the benefits of the LDL factorization for large-scale differential matrix equation solvers, Linear Algebra and its Applications, 480 (2015), 44-71. [29] I. Lasiecka, Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel, S. Piazzera), Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1855 (2004), 313-369. [30] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories Ⅱ. Abstract Hyperbolic-like Systems over a Finite Time Horizon, Cambridge University Press, Cambridge, UK, 2000. [31] I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by $C_{0}$-semigroups satisfying a singular estimate, J. Optim. Theory Appl., 136 (2008), 229-246. [32] T. Levajković and H. Mena, On deterministic and stochastic linear quadratic control problem, in Current Trends in Analysis and Its Applications. Trends in Mathematics. (eds. V. Mityushev, M. Ruzhansky), Birkhäuser, Cham, (2015), 315-322. [33] T. Levajković, H. Mena and A. Tuffaha, The stochastic linear quadratic control problem: A chaos expansion approach, Evolution Equations and Control Theory, 5 (2016), 105-134. [34] T. Levajković, H. Mena and A. Tuffaha, A numerical approximation framework for the stochastic linear quadratic regulator problem on Hilbert spaces, Applied Mathematics and Optimization, 75 (2017), 499-523. [35] V. Mehrmann, The Autonomous Linear Quadratic Control Problem, Springer-Verlag, Berlin, 1991. [36] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [37] I. Petersen, V. Ugrinovskii and A. Savkin, Robust Control Design Using H∞ Methods, Springer-Verlag, London, 2000.

show all references

##### References:
 [1] H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser, Basel, Switzerland, 2003. [2] A. H. Al-Mohy and N. J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput., 33 (2011), 488-511. [3] E. Arias, V. Hernández, J. Ibanes and J. Peinado, A family of BDF algorithms for solving differential matrix Riccati equations using adaptive techniques, Procedia Computer Science, 1 (2010), 2569-2577. [4] E. Armstrong, An extension of Bass' algorithm for stabilizing linear continuous constant systems, IEEE Trans. Automatic Control, AC-20 (1975), 153-154. [5] A. Balakrishnan, Applied Functional Analysis, Springer-Verlag, New York, 1981. [6] H. Banks, R. Smith and Y. Wang, The modeling of piezoceramic patch interactions with shells, plates and beams, Quart. Appl. Math., 53 (1995), 353-381. [7] A. Bátkai, P. Csomós, B. Farkas and G. Nickel, Operator splitting for non-autonomous evolution equations, J. Funct. Anal., 260 (2011), 2163-2192. [8] A. Bátkai, P. Csomós and G. Nickel, Operator splittings and spatial approximations for evolution equations, J. Evol. Eqs., 9 (2009), 613-636. [9] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, 1993. [10] P. Benner, P. Ezzatti, H. Mena, E. S. Quintana-Ortí and A. Remón, Solving matrix equations on multi-core and many-core architectures, Algorithms, 6 (2013), 857-870. [11] P. Benner and H. Mena, Numerical solution of the infinite-dimensional LQR-problem and the associated differential Riccati equations, Journal of Numerical Mathematics (2016), in press. [12] P. Benner and H. Mena, Rosenbrock methods for solving differential Riccati equations, IEEE Transactions on Automatic Control, 58 (2013), 2950-2957. [13] P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM Mitteilungen, 36 (2013), 32-52. [14] P. Csomós and J. Winckler, A semigroup proof for the well-posedness of the linearised shallow water equations, J. Anal. Math., 43 (2017), 445-459. [15] G. Da Prato, Direct solution of a Riccati equation arising in stochastic control theory, Appl. Math. Optim., 11 (1984), 191-208. [16] G. Da Prato, P. Kunstmann, I. Lasiecka, A. Lunardi, R. Schnaubelt and L. Weis, Functional Analytic Methods for Evolution Equations, Springer-Verlag, Berlin, 2004. [17] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. [18] F. Flandoli, Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary, Appl. Math. Optim., 14 (1986), 107-129. [19] C. Hafizoglu, I. Lasiecka, T. Levajković, H. Mena and A. Tuffaha, The stochastic linear quadratic problem with singular estimates, SIAM J. Control Optim., 55 (2017), 595-626. [20] E. Hansen and A. Ostermann, Exponential splitting for unbounded operators, Math. Comput., 78 (2009), 1485-1496. [21] A. Ichikawa, Dynamic programming approach to stochastic evolution equation, SIAM J. Control. Optim., 17 (1979), 152-174. [22] A. Ichikawa and H. Katayama, Remarks on the time-varying H∞ Riccati equations, Sys. Cont. Lett., 37 (1999), 335-345. [23] O. Iftime and M. Opmeer, A representation of all bounded selfadjoint solutions of the algebraic Riccati equation for systems with an unbounded observation operator, Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 14-07 (2004), 2865-2870. [24] K. Ito and F. Kappel, Evolution Equations and Approximations, World Scientific, Singapore, 2002. [25] T. Jahnke and Ch. Lubich, Error bounds for exponential operator splittings, BIT, 40 (2000), 735-744. [26] D. Kleinman, On an iterative technique for Riccati equation computations, IEEE Trans. Automatic Control, AC-13 (1968), 114-115. [27] A. Kofler, H. Mena and A. Ostermann, Splitting methods for stochastic partial differential equations, preprint [28] N. Lang, H. Mena and J. Saak, On the benefits of the LDL factorization for large-scale differential matrix equation solvers, Linear Algebra and its Applications, 480 (2015), 44-71. [29] I. Lasiecka, Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel, S. Piazzera), Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1855 (2004), 313-369. [30] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories Ⅱ. Abstract Hyperbolic-like Systems over a Finite Time Horizon, Cambridge University Press, Cambridge, UK, 2000. [31] I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by $C_{0}$-semigroups satisfying a singular estimate, J. Optim. Theory Appl., 136 (2008), 229-246. [32] T. Levajković and H. Mena, On deterministic and stochastic linear quadratic control problem, in Current Trends in Analysis and Its Applications. Trends in Mathematics. (eds. V. Mityushev, M. Ruzhansky), Birkhäuser, Cham, (2015), 315-322. [33] T. Levajković, H. Mena and A. Tuffaha, The stochastic linear quadratic control problem: A chaos expansion approach, Evolution Equations and Control Theory, 5 (2016), 105-134. [34] T. Levajković, H. Mena and A. Tuffaha, A numerical approximation framework for the stochastic linear quadratic regulator problem on Hilbert spaces, Applied Mathematics and Optimization, 75 (2017), 499-523. [35] V. Mehrmann, The Autonomous Linear Quadratic Control Problem, Springer-Verlag, Berlin, 1991. [36] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [37] I. Petersen, V. Ugrinovskii and A. Savkin, Robust Control Design Using H∞ Methods, Springer-Verlag, London, 2000.
Solution $w(t, \xi)$ to the one-dimensional advection equation (7) without control ($B = 0$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$
Solution $w(t, \xi)$ to the one-dimensional advection equation (7) without control ($B = 0$) at time $t = 3.2$ by using Lax-Wendroff scheme (right panel) with time step $\tau = 10^{-3}$
Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) without control ($B = 0$) by using Godunov's scheme or Lax-Wendroff scheme with time step $\tau = 10^{-3}$
Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$
Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 3.2$ by using Lax-Wendroff scheme with time step $\tau = 10^{-3}$
Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 3.2$ by using Godunov's scheme or Lax-Wendroff scheme with time step $\tau = 10^{-3}$
Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$
Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Lax-Wendroff scheme with time step $\tau = 10^{-3}$
Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$
Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Lax-Wendroff scheme with time step $\tau = 10^{-3}$
Enlargements of Figure 9
Enlargements of Figure 10
Solution to two-dimensional advection equation (8) at time $t = 3$ without control ($B = 0$) by using Fourier's method (left column) and Lax-Wendroff scheme (right column) with time step $\tau = 10^{-3}$ and number of grid points $N_\xi = N_\eta = 32, 64,128$ from top to bottom, respectively
Solution to two-dimensional advection equation (8) at time $t = 3$ with distributed control ($B = \text{I}_{\mathcal U}$) by using Lax-Wendroff scheme (left column) and Fourier-Splitting (right column) with time step $\tau = 10^{-3}$ and number of grid points $N_\xi = N_\eta = 32, 64$ from top to bottom, respectively
Volume ratio $\mathcal V_2(t)$ of two-dimensional advection equation (8) with distributed control ($B = \text{I}_{\mathcal U}$) by using Lax-Wendroff scheme and Fourier-Splitting with time step $\tau = 10^{-3}$
Solution to the one-dimensional linearized shallow water equations (9) without control ($B = 0$) at time $t = 25$ with time step $\tau = 10^{-4}$
Time-evolution of the volume ratio $\mathcal V_3$ for the one-dimensional linearized shallow water equations (9) without control ($B = 0$) at time $t = 25$ with time step $\tau = 10^{-4}$
Solution to the one-dimensional linearized shallow water equations (9) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 25$ with time step $\tau = 10^{-4}$
Time-evolution of the volume ratio $\mathcal V_3$ for the one-dimensional linearized shallow water equations (9) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 25$ with time step $\tau = 10^{-4}$
Solution to the one-dimensional linearized shallow water equations (9) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 40$ with time step $\tau = 10^{-4}$
Time-evolution of the volume ratio $\mathcal V_3$ for the one-dimensional linearized shallow water equations (9) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 40$ with time step $\tau = 10^{-4}$
Solution to two-dimensional shallow water equations (12) at time $t = 4.5$ without control ($B = 0$) by using Fourier's method (left column) and Lax-Wendroff scheme (right column) with time step $\tau = 10^{-4}$ and number of grid points $N_\xi = N_\eta = 16, 32, 64$ from top to bottom, respectively
Solution to two-dimensional shallow water equations (12) at time $t = 4.5$ with distributed control ($B = \text{I}_{\mathcal U}$) by using Lax-Wendroff scheme (left column) and Fourier-Splitting (right column) with time step $\tau = 10^{-4}$ and number of grid points $N_\xi = N_\eta = 16, 32$ from top to bottom, respectively
Volume ratio $\mathcal V_4$ for two-dimensional shallow water equations (12) for distributed control ($B = \text{I}_{\mathcal U}$) by using time step $\tau = 10^{-4}$
Solution to two-dimensional shallow water equations (12) at time $t = 4.5$ with sink-like control ($B = \text{I}_{\Gamma_{2\ell}}$) by using Lax-Wendroff scheme (left column) and Fourier-Splitting (right column) with time step $\tau = 10^{-4}$ and number of grid points $N_\xi = N_\eta = 16, 32$ from top to bottom, respectively
Volume ratio $\mathcal V_4$ for two-dimensional shallow water equations (12) for the sink-like control ($B = \text{I}_{\Gamma_1}$)
Two-dimensional linearized shallow water equations with control matrix $B = \Gamma_{2r}$ representing a sink. Comparison between no control (left column), sequential splitting (column in the middle) and Strang splitting (right column)
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