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Receding horizon control for the stabilization of the wave equation

  • * Corresponding author: Behzad Azmi

    * Corresponding author: Behzad Azmi 
This work has been supported by the International Research Training Group IGDK1754, funded by the DFG and FWF, and the ERC advanced grant 668998 (OCLOC) under the EU's H2020 research program
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  • Stabilization of the wave equation by the receding horizon framework is investigated. Distributed control, Dirichlet boundary control, and Neumann boundary control are considered. Moreover for each of these control actions, the well-posedness of the control system and the exponential stability of Receding Horizon Control (RHC) with respect to a proper functional analytic setting are investigated. Observability conditions are necessary to show the suboptimality and exponential stability of RHC. Numerical experiments are given to illustrate the theoretical results.

    Mathematics Subject Classification: Primary: 49N35, 93C20, 93D20.

    Citation:

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  • Figure 2.  Control domains

    Figure 1.  Snapshots of the uncontrolled state corresponding to Example 5.1

    Figure 3.  Evolution of $L^2(\omega)$-norm for RHC corresponding to Example 5.1 with different prediction horizons $T$

    Figure 4.  Evolution of $\|\mathcal{Y}_{rh}(t)\|_{\mathcal{H}}$ for different choices of $T$

    Figure 5.  Snapshots of receding horizon state for the choice of $T = 1.5$ corresponding to Example 5.1

    Figure 6.  Snapshots of receding horizon state for the choice of $T = 1.5$ corresponding to Example 5.2

    Figure 7.  Numerical results corresponding to Example 5.3

    Algorithm 1 Receding Horizon Algorithm
    Require: Let the prediction horizon $T$, the sampling time $\delta<T$, and the initial point $(y^1_0, y^2_0)\in \mathcal{H}$ be given. Then we proceed through the following steps:
    1: $k := 0, t_0 := 0$ and $\mathcal{Y}_{rh}(t_0):=(y^1_0, y^2_0)$.
    2: Find the optimal pair $(\mathcal{Y}_T^*(\cdot;\mathcal{Y}_{rh}(t_k), t_k), u^*_T(\cdot;\mathcal{Y}_{rh}(t_k), t_k))$ over the time horizon $[t_k, t_k +T]$ by solving the finite horizon open-loop problem
    $ \begin{split} &\min_{u\in L^2(t_k, t_k+T; \mathcal{U})}J_T(u; \mathcal{Y}_{rh}(t_k)):= \min_{u\in L^2(t_k, t_k+T; \mathcal{U})} \int^{t_k+T}_{t_k} \ell(\mathcal{Y}(t), u(t))dt, \\ \text{ s.t } & \begin{cases} \dot{\mathcal{Y}} = \mathcal{A}\mathcal{Y}+\mathcal{B}u t \in (t_k, t_k+T) \\ \mathcal{Y}(t_k) = \mathcal{Y}_{rh}(t_k) \end{cases} \end{split} $
    3: Set
    $ \begin{split} u_{rh}(\tau)&:=u^*_T(\tau; y_{rh}(t_k), t_k) \text{ for all } \tau \in [t_k, t_k+\delta), \\ \mathcal{Y}_{rh}(\tau)&:=\mathcal{Y}^*_T(\tau; y_{rh}(t_k), t_k) \text{ for all } \tau \in [t_k, t_k+\delta], \\ t_{k+1} &:= t_k +\delta, \\ k &:= k+1. \end{split} $
    4: Go to step 2.
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    Algorithm 2 RHC($\mathcal{Y}_0, T_{\infty}$)
    Input: Let a final computational time horizon $T_{\infty}$, and an initial state $\mathcal{Y}_0:=(y^1_0, y^2_0) \in \mathcal{H}$ be given.
    1: Choose a prediction horizon $T < T_{\infty}$ and a sampling time $\delta \in (0, T]$.
    2: Consider a grid $0 = t_0 < t_1 < \cdots <t_r = T_{\infty}$ on the interval $[0, T_{\infty}]$ where $t_i = i\delta$ for $i = 0, \dots, r$.
    3: for $i = 0, \dots, r-1$ do
    Solve the open-loop subproblem on $[t_i, t_i + T]$
    $ \begin{split} \min & \frac{1}{2}\int_{t_i}^{t_i + T} \|\mathcal{Y}(t)\|^2_{\mathcal{H}}dt + \frac{\beta}{2}\int_{t_i}^{t_i + T}\|u(t)\|^2_{\mathcal{U}}dt, \\ \text{ subject to } & \begin{cases} \dot{\mathcal{Y}} = \mathcal{A}\mathcal{Y}+\mathcal{B}u t \in (t_k, t_k+T), \\ \mathcal{Y}(t_i) = \mathcal{Y}_T^*(t_i) \mbox{ if } i \geq 1 \mbox{ or } \mathcal{Y}(t_i) = (y^1_0, y^2_0) \mbox{ if } i = 0, \end{cases} \end{split} $
    where $\mathcal{Y}_T^*(\cdot)$ is the solution to the previous subproblem on $[t_{i-1}, t_{i-1}+T]$.
    4: The model predictive pair $\left( \mathcal{Y}_{rh}^*(\cdot), u_{rh}^*(\cdot)\right)$ is the concatenation of the optimal pairs $\left( \mathcal{Y}_T^*(\cdot), u_T^*(\cdot)\right)$ on the finite horizon intervals $[t_i, t_{i+1}]$ with $i = 0, \dots, r-1$.
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    Table 1.  Numerical results for Example 5.1

    Prediction Horizon $J_{T_{\infty}}$ $\|\mathcal{Y}_{rh}\|_{L^2(0, T_{\infty};\mathcal{H}_1)}$ $\|\mathcal{Y}_{rh}(T_{\infty})\|_{\mathcal{H}_1}$iter
    $T = 1.5$ $8.20\times 10^2$$ 40.19$$ 2.62 \times 10^{-8}$ $1515$
    $T = 1$$1.13\times 10^3$$47.40$$ 3.03\times10^{-6}$ $847$
    $T = 0.5$ $3.13\times 10^{3}$$79.10$ $2.00 \times 10^{-3}$$550$
    $T = 0.25$$1. 94\times 10^{4}$$197.43$ $3.79 \times 10^{-1}$$373$
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    Table 2.  Numerical results for Example 5.2

    Prediction Horizon $J_{T_{\infty}}$ $\|\mathcal{Y}_{rh}\|_{L^2(0, T_{\infty};\mathcal{H}_2)}$ $\|\mathcal{Y}_{rh}(T_{\infty})\|_{\mathcal{H}_2}$iter
    $T = 1.5$ $2.20$$ 1.93$$2.11 \times 10^{-6}$ $715$
    $T = 1$$2.75$$2.23$$ 3.42\times10^{-5}$ $599$
    $T = 0.5$ $6.77$$3.64$ $6.00 \times 10^{-3}$$445$
    $T = 0.25$$33.75$$8.20$ $2.36\times 10^{-1}$$359$
     | Show Table
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    Table 3.  Numerical results for Example 5.3

    Prediction Horizon $J_{T_{\infty}}$ $\|\mathcal{Y}_{rh}\|_{L^2(0, T_{\infty};\mathcal{H}_3)}$ $\|\mathcal{Y}_{rh}(T_{\infty})\|_{\mathcal{H}_3}$iter
    $T = 1.5$ $1.30 \times 10^{4}$$161.47$$3.85 \times 10^{-6}$ $5348$
    $T = 1$$1.67 \times 10^{4}$$182.97$$7.08\times10^{-5}$ $3303$
    $T = 0.5$ $3.92 \times 10^{4}$$280.22$ $4.91 \times 10^{-2}$$1507$
    $T = 0.25$$2.41 \times 10^{5}$$694.40$ $9.26$$823$
     | Show Table
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