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Fourier-splitting method for solving hyperbolic LQR problems

  • * Corresponding author: P. Csomós

    * Corresponding author: P. Csomós 
The first author is supported by the National Research, Development and Innovation Fund (Hungary) under the grant PD121117.
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  • 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.

    Mathematics Subject Classification: Primary: 35Q93, 49J20; Secondary: 65M22.

    Citation:

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  • Figure 1.  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 $\tau = 10^{-3}$

    Figure 2.  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 $\tau = 10^{-3}$

    Figure 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 $\tau = 10^{-3}$

    Figure 4.  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 $\tau = 10^{-3}$

    Figure 5.  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 $\tau = 10^{-3}$

    Figure 6.  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 $\tau = 10^{-3}$

    Figure 7.  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 $\tau = 10^{-3}$

    Figure 8.  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 $\tau = 10^{-3}$

    Figure 9.  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 $\tau = 10^{-3}$

    Figure 10.  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 $\tau = 10^{-3}$

    Figure 11.  Enlargements of Figure 9

    Figure 12.  Enlargements of Figure 10

    Figure 13.  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

    Figure 14.  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

    Figure 15.  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 $\tau = 10^{-3}$

    Figure 16.  Solution to the one-dimensional linearized shallow water equations (9) without control ($B = 0$) at time $t = 25$ with time step $\tau = 10^{-4}$

    Figure 17.  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 $\tau = 10^{-4}$

    Figure 18.  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 $\tau = 10^{-4}$

    Figure 19.  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 $\tau = 10^{-4}$

    Figure 20.  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 $\tau = 10^{-4}$

    Figure 21.  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 $\tau = 10^{-4}$

    Figure 22.  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

    Figure 23.  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

    Figure 24.  Volume ratio $\mathcal V_4$ for two-dimensional shallow water equations (12) for distributed control ($B = \text{I}_{\mathcal U}$) by using time step $\tau = 10^{-4}$

    Figure 25.  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

    Figure 26.  Volume ratio $\mathcal V_4$ for two-dimensional shallow water equations (12) for the sink-like control ($B = \text{I}_{\Gamma_1}$)

    Figure 27.  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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