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doi: 10.3934/mcrf.2022006
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The discretized backstepping method: An application to a general system of $2\times 2$ linear balance laws

 Institut de mathematiques de Toulouse, 118 route de Narbonne, Toulouse, France

Received  July 2021 Revised  January 2022 Early access March 2022

In this paper, we introduce the numerical backstepping method by applying it to a problem of finite-time stabilization for a system of $2 \times 2$ balance laws discretized thanks to the upwind scheme. On the one hand, we illustrate on an example that the scheme used to compute the feedback control cannot be chosen arbitrarily. On the other hand, an algorithm is given to construct this control properly and an approached finite-time stabilization result is proven.

Citation: Mathias Dus. The discretized backstepping method: An application to a general system of $2\times 2$ linear balance laws. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022006
References:
 [1] J. Auriol, Robust Design of Backstepping Controllers for Systems of Linearhyperbolic PDEs, PhD thesis, Mines ParisTech, Paris, 2018. [2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099. [3] G. Bastin and J.-M. Coron, On boundary feedback stabilization of non-uniform linear $2\times2$ hyperbolic systems over a bounded interval, Systems Control Lett., 60 (2011), 900-906.  doi: 10.1016/j.sysconle.2011.07.008. [4] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-32062-5. [5] G. Bastin, J.-M. Coron and B. d'Andréa Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187.  doi: 10.3934/nhm.2009.4.177. [6] J.-M. Coron, On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim., 37 (1999), 1874-1896.  doi: 10.1137/S036301299834140X. [7] J.-M. Coron, R. Vazquez, M. Krstic and G. Bastin, Local exponential $H^2$ stabilization of a $2\times 2$ quasilinear hyperbolic system using backstepping, SIAM J. Control Optim., 51 (2013), 2005-2035.  doi: 10.1137/120875739. [8] J. de Halleux, C. Prieur, J.-M. Coron, B. Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.  doi: 10.1016/S0005-1098(03)00109-2. [9] F. Di Meglio, Dynamic and Control of Slugging in Oil Production, PhD thesis, Mines ParisTech, Paris, 2011. [10] F. Di Meglio, G. Kaasa, N. Petit and V. Alstad, Slugging in multiphase flow as a mixed initial-boundary value problem for a quasilinear hyperbolic system, In Proceedings of the 2011 American Control Conference, (2011), 3589–3596. [11] F. Di Meglio, R. Vazquez and M. Krstic, Stabilization of a system of $n+1$ coupled first-order hyperbolic linear pdes with a single boundary input, IEEE Trans. Automat. Control, 58 (2013), 3097-3111.  doi: 10.1109/TAC.2013.2274723. [12] A. Diagne, G. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030. [13] S. Dudret, K. Beauchard, F. Ammouri and P. Rouchon, Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models, In 2012 American Control Conference (ACC), (2012), 3352–3358. [14] M. Dus, Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition, Mathematics of Control, Signals, and Systems, 34 (2022), 37-65.  doi: 10.1007/s00498-021-00301-2. [15] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, In Handb. Numer. Anal., (2000), 713–1020. [16] A. Hayat, Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the $C^1$ norm, SIAM J. Control Optim., 57 (2019), 3603-3638.  doi: 10.1137/17M1150803. [17] A. Hayat, On boundary stability of inhomogeneous $2\times2$ 1-D hyperbolic systems for the $C^1$ norm, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 82, 31 pp. doi: 10.1051/cocv/2018059. [18] J. M. Holte, Discrete Gronwall Lemma and Applications, Technical report, MAA north central section meeting at und, 2009. [19] L. Hu, F. Di Meglio, R. Vazquez and M. Krstic, Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 3301-3314.  doi: 10.1109/TAC.2015.2512847. [20] L. Hu and G. Olive, Null controllability and finite-time stabilization in minimal time of one-dimensional first-order 2 $\times$ 2 linear hyperbolic systems, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 96, 18 pp. doi: 10.1051/cocv/2021091. [21] M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, 1$^{st}$ edition, John Wiley and Sons, Inc., USA, 1995. [22] M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005. [23] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs, A course on backstepping designs. Advances in Design and Control, 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607. [24] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253. [25] A. Smyshlyaev, B. Guo and M. Krstic, Arbitrary decay rate for euler-bernoulli beam by backstepping boundary feedback, IEEE Trans. Automat. Control, 54 (2009), 1134-1140.  doi: 10.1109/TAC.2009.2013038. [26] A. Smyshlyaev and M. Krstic, Adaptive Control of Parabolic PDEs, Princeton University Press, Princeton, NJ, 2010.  doi: 10.1515/9781400835362. [27] R. Vazquez and M. Krstic, Marcum $Q$-functions and explicit kernels for stabilization of $2\times2$ linear hyperbolic systems with constant coefficients, Systems Control Lett., 68 (2014), 33-42.  doi: 10.1016/j.sysconle.2014.02.008. [28] C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM Control Optim. Calc. Var., 7 (2002), 421-442.  doi: 10.1051/cocv:2002062.

show all references

References:
 [1] J. Auriol, Robust Design of Backstepping Controllers for Systems of Linearhyperbolic PDEs, PhD thesis, Mines ParisTech, Paris, 2018. [2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099. [3] G. Bastin and J.-M. Coron, On boundary feedback stabilization of non-uniform linear $2\times2$ hyperbolic systems over a bounded interval, Systems Control Lett., 60 (2011), 900-906.  doi: 10.1016/j.sysconle.2011.07.008. [4] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-32062-5. [5] G. Bastin, J.-M. Coron and B. d'Andréa Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187.  doi: 10.3934/nhm.2009.4.177. [6] J.-M. Coron, On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim., 37 (1999), 1874-1896.  doi: 10.1137/S036301299834140X. [7] J.-M. Coron, R. Vazquez, M. Krstic and G. Bastin, Local exponential $H^2$ stabilization of a $2\times 2$ quasilinear hyperbolic system using backstepping, SIAM J. Control Optim., 51 (2013), 2005-2035.  doi: 10.1137/120875739. [8] J. de Halleux, C. Prieur, J.-M. Coron, B. Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.  doi: 10.1016/S0005-1098(03)00109-2. [9] F. Di Meglio, Dynamic and Control of Slugging in Oil Production, PhD thesis, Mines ParisTech, Paris, 2011. [10] F. Di Meglio, G. Kaasa, N. Petit and V. Alstad, Slugging in multiphase flow as a mixed initial-boundary value problem for a quasilinear hyperbolic system, In Proceedings of the 2011 American Control Conference, (2011), 3589–3596. [11] F. Di Meglio, R. Vazquez and M. Krstic, Stabilization of a system of $n+1$ coupled first-order hyperbolic linear pdes with a single boundary input, IEEE Trans. Automat. Control, 58 (2013), 3097-3111.  doi: 10.1109/TAC.2013.2274723. [12] A. Diagne, G. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030. [13] S. Dudret, K. Beauchard, F. Ammouri and P. Rouchon, Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models, In 2012 American Control Conference (ACC), (2012), 3352–3358. [14] M. Dus, Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition, Mathematics of Control, Signals, and Systems, 34 (2022), 37-65.  doi: 10.1007/s00498-021-00301-2. [15] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, In Handb. Numer. Anal., (2000), 713–1020. [16] A. Hayat, Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the $C^1$ norm, SIAM J. Control Optim., 57 (2019), 3603-3638.  doi: 10.1137/17M1150803. [17] A. Hayat, On boundary stability of inhomogeneous $2\times2$ 1-D hyperbolic systems for the $C^1$ norm, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 82, 31 pp. doi: 10.1051/cocv/2018059. [18] J. M. Holte, Discrete Gronwall Lemma and Applications, Technical report, MAA north central section meeting at und, 2009. [19] L. Hu, F. Di Meglio, R. Vazquez and M. Krstic, Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 3301-3314.  doi: 10.1109/TAC.2015.2512847. [20] L. Hu and G. Olive, Null controllability and finite-time stabilization in minimal time of one-dimensional first-order 2 $\times$ 2 linear hyperbolic systems, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 96, 18 pp. doi: 10.1051/cocv/2021091. [21] M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, 1$^{st}$ edition, John Wiley and Sons, Inc., USA, 1995. [22] M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005. [23] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs, A course on backstepping designs. Advances in Design and Control, 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607. [24] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253. [25] A. Smyshlyaev, B. Guo and M. Krstic, Arbitrary decay rate for euler-bernoulli beam by backstepping boundary feedback, IEEE Trans. Automat. Control, 54 (2009), 1134-1140.  doi: 10.1109/TAC.2009.2013038. [26] A. Smyshlyaev and M. Krstic, Adaptive Control of Parabolic PDEs, Princeton University Press, Princeton, NJ, 2010.  doi: 10.1515/9781400835362. [27] R. Vazquez and M. Krstic, Marcum $Q$-functions and explicit kernels for stabilization of $2\times2$ linear hyperbolic systems with constant coefficients, Systems Control Lett., 68 (2014), 33-42.  doi: 10.1016/j.sysconle.2014.02.008. [28] C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM Control Optim. Calc. Var., 7 (2002), 421-442.  doi: 10.1051/cocv:2002062.
The domain where $P$ is defined
The grid for the computation of $P$
The $L^2$ norm of the solution for case 1
Spectrums of discretized operators for case 1
The $L^2$ norm of the solution for case 2
Spectra of discretized operators for case 2
The space grids for $\alpha = 2$
The definition of $\Pi_{f \leftarrow c}$
The definition of $\Pi_{c \leftarrow f}$
The non zero coefficients for $\Gamma_{11}$ ($\alpha = 3$)
The non zero coefficients for $\Gamma_{12}$ ($\alpha = 3$)
The kernels of the closed-loop operator for the naive method
The spectrum of the closed-loop operator for the naive method
The kernels of the closed-loop operator for less naive method
The spectrum of the closed-loop operator for the less naive method
The $L^2$ norm of the solution (log10 scale)
When the perturbation term is non zero
When the perturbation term is zero
When velocities are different
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