February  2018, 38(2): 449-484. doi: 10.3934/dcds.2018021

Receding horizon control for the stabilization of the wave equation

1. 

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraβe 69, A-4040 Linz, Austria

2. 

Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria

* Corresponding author: Behzad Azmi

Received  March 2017 Revised  August 2017 Published  February 2018

Fund Project: 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

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.

Citation: Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021
References:
[1]

K. Ammari, Dirichlet boundary stabilization of the wave equation, Asymptot. Anal., 30 (2002), 117-130.

[2]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386 (electronic). doi: 10.1051/cocv:2001114.

[3]

B. Azmi and K. Kunisch, On the stabilizability of the Burgers equation by receding horizon control, SIAM J. Control Optim., 54 (2016), 1378-1405. doi: 10.1137/15M1030352.

[4]

L. Bales and I. Lasiecka, Continuous finite elements in space and time for the nonhomogeneous wave equation, Comput. Math. Appl., 27 (1994), 91-102. doi: 10.1016/0898-1221(94)90048-5.

[5]

——, Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous $L_2$ Dirichlet boundary data, Math. Comp., 64 (1995), 89-115 doi: 10.2307/2153324.

[6]

W. BangerthM. Geiger and R. Rannacher, Adaptive Galerkin finite element methods for the wave equation, Comput. Methods Appl. Math., 10 (2010), 3-48. doi: 10.2478/cmam-2010-0001.

[7]

W. Bangerth and R. Rannacher, Adaptive finite element techniques for the acoustic wave equation, J. Comput. Acoust., 9 (2001), 575-591. doi: 10.1142/S0218396X01000668.

[8]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[9]

J. Barzilai and J. M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.

[10]

R. BeckerD. Meidner and B. Vexler, Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Methods Softw., 22 (2007), 813-833. doi: 10.1080/10556780701228532.

[11]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[12]

G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM Journal on Control and Optimization, 17 (1979), 66-81. doi: 10.1137/0317007.

[13]

H. Chen and F. Allgöwer, A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica J. IFAC, 34 (1998), 1205-1217. doi: 10.1016/S0005-1098(98)00073-9.

[14]

N. Cîndea and A. Münch, A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations, Calcolo, 52 (2015), 245-288. doi: 10.1007/s10092-014-0116-x.

[15]

J. -M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007.

[16]

R. Curtain and K. Morris, Transfer functions of distributed parameter systems: A tutorial, Automatica J. IFAC, 45 (2009), 1101-1116. doi: 10.1016/j.automatica.2009.01.008.

[17]

Y.-H. Dai and H. Zhang, Adaptive two-point stepsize gradient algorithm, Numer. Algorithms, 27 (2001), 377-385. doi: 10.1023/A:1013844413130.

[18]

A. Ern and J. -L. Guermond, Theory and Practice of Finite Elements, vol. 159 of Applied Mathematical Sciences, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[19]

F. Flandoli, Invertibility of Riccati operators and controllability of related systems, Systems Control Lett., 9 (1987), 65-72. doi: 10.1016/0167-6911(87)90010-7.

[20]

G. GrimmM. J. MessinaS. E. Tuna and A. R. Teel, Model predictive control: For want of a local control Lyapunov function, all is not lost, IEEE Trans. Automat. Control, 50 (2005), 546-558. doi: 10.1109/TAC.2005.847055.

[21]

L. Grüne, Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems, SIAM J. Control Optim., 48 (2009), 1206-1228. doi: 10.1137/070707853.

[22]

L. Grüne and A. Rantzer, On the infinite horizon performance of receding horizon controllers, IEEE Trans. Automat. Control, 53 (2008), 2100-2111. doi: 10.1109/TAC.2008.927799.

[23]

M. Gugat, Exponential stabilization of the wave equation by Dirichlet integral feedback, SIAM J. Control Optim., 53 (2015), 526-546. doi: 10.1137/140977023.

[24]

M. GugatG. Leugering and G. Sklyar, $L^p$-optimal boundary control for the wave equation, SIAM J. Control Optim., 44 (2005), 49-74. doi: 10.1137/S0363012903419212.

[25]

M. GugatE. Trélat and E. Zuazua, Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property, Systems Control Lett., 90 (2016), 61-70. doi: 10.1016/j.sysconle.2016.02.001.

[26]

W. Guo and B.-Z. Guo, Adaptive output feedback stabilization for one-dimensional wave equation with corrupted observation by harmonic disturbance, SIAM J. Control Optim., 51 (2013), 1679-1706. doi: 10.1137/120873212.

[27]

E. Hendrickson and I. Lasiecka, Numerical approximations of solutions to Riccati equations arising in boundary control problems for the wave equation, in Optimal control of differential equations (Athens, OH, 1993), vol. 160 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1994,111-132.

[28]

K. Ito and K. Kunisch, Receding horizon optimal control for infinite dimensional systems, ESAIM Control Optim. Calc. Var., 8 (2002), 741-760 (electronic). A tribute to J. L. Lions. doi: 10.1051/cocv:2002032.

[29]

A. Jadbabaie and J. Hauser, On the stability of receding horizon control with a general terminal cost, IEEE Trans. Automat. Control, 50 (2005), 674-678. doi: 10.1109/TAC.2005.846597.

[30]

C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 107 (1993), 117-129. doi: 10.1016/0045-7825(93)90170-3.

[31]

O. Karakashian and C. Makridakis, Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations, Math. Comp., 74 (2005), 85-102. doi: 10.1090/S0025-5718-04-01654-0.

[32]

A. KrönerK. Kunisch and B. Vexler, Semismooth Newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49 (2011), 830-858. doi: 10.1137/090766541.

[33]

K. KunischP. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244. doi: 10.1137/141001366.

[34]

J. E. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256. doi: 10.1137/0326068.

[35]

I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2(0, ∞; L_2(Γ))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390. doi: 10.1016/0022-0396(87)90025-8.

[36]

——, Sharp regularity theory for second order hyperbolic equations of Neumann type. I. L2 nonhomogeneous data, Ann. Mat. Pura Appl. (4), 157 (1990), 285-367. doi: 10.1007/BF01765322.

[37]

——, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, vol. 164 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0006880.

[38]

——, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224. doi: 10.1007/BF01182480.

[39]

——, Algebraic Riccati equations arising from systems with unbounded input-solution operator: applications to boundary control problems for wave and plate equations, Nonlinear Anal., 20 (1993), 659-695. doi: 10.1016/0362-546X(93)90026-O.

[40]

J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.

[41]

——, Contrôle des Systémes Distribués Singuliers, vol. 13 of Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Gauthier-Villars, Montrouge, 1983.

[42]

J. -L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.

[43]

——, Non-homogeneous Boundary Value Problems and Applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182.

[44]

D. Q. MayneJ. B. RawlingsC. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica J. IFAC, 36 (2000), 789-814. doi: 10.1016/S0005-1098(99)00214-9.

[45]

B. S. Mordukhovich and J. -P. Raymond, Neumann boundary control of hyperbolic equations with pointwise state constraints, SIAM J. Control Optim., 43 (2004/05), 1354-1372 (electronic). doi: 10.1137/S0363012903431177.

[46]

B. S. Mordukhovich and J.-P. Raymond, Dirichlet boundary control of hyperbolic equations in the presence of state constraints, Appl. Math. Optim., 49 (2004), 145-157. doi: 10.1007/BF02638149.

[47]

A. Münch and A. F. Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation, ESAIM Control Optim. Calc. Var., 13 (2007), 265-293 (electronic). doi: 10.1051/cocv:2007009.

[48]

M. Reble and F. Allgöwer, Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems, Automatica J. IFAC, 48 (2012), 1812-1817. doi: 10.1016/j.automatica.2012.05.067.

[49]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[50]

L. T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite difference space discretization of the $1-d$ wave equation, Adv. Comput. Math., 26 (2007), 337-365. doi: 10.1007/s10444-004-7629-9.

[51]

R. Triggiani, Exact boundary controllability on $L_2(Ω)× H^{-1}(Ω)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partialΩ$, and related problems, Appl. Math. Optim., 18 (1988), 241-277. doi: 10.1007/BF01443625.

show all references

References:
[1]

K. Ammari, Dirichlet boundary stabilization of the wave equation, Asymptot. Anal., 30 (2002), 117-130.

[2]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386 (electronic). doi: 10.1051/cocv:2001114.

[3]

B. Azmi and K. Kunisch, On the stabilizability of the Burgers equation by receding horizon control, SIAM J. Control Optim., 54 (2016), 1378-1405. doi: 10.1137/15M1030352.

[4]

L. Bales and I. Lasiecka, Continuous finite elements in space and time for the nonhomogeneous wave equation, Comput. Math. Appl., 27 (1994), 91-102. doi: 10.1016/0898-1221(94)90048-5.

[5]

——, Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous $L_2$ Dirichlet boundary data, Math. Comp., 64 (1995), 89-115 doi: 10.2307/2153324.

[6]

W. BangerthM. Geiger and R. Rannacher, Adaptive Galerkin finite element methods for the wave equation, Comput. Methods Appl. Math., 10 (2010), 3-48. doi: 10.2478/cmam-2010-0001.

[7]

W. Bangerth and R. Rannacher, Adaptive finite element techniques for the acoustic wave equation, J. Comput. Acoust., 9 (2001), 575-591. doi: 10.1142/S0218396X01000668.

[8]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[9]

J. Barzilai and J. M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.

[10]

R. BeckerD. Meidner and B. Vexler, Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Methods Softw., 22 (2007), 813-833. doi: 10.1080/10556780701228532.

[11]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[12]

G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM Journal on Control and Optimization, 17 (1979), 66-81. doi: 10.1137/0317007.

[13]

H. Chen and F. Allgöwer, A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica J. IFAC, 34 (1998), 1205-1217. doi: 10.1016/S0005-1098(98)00073-9.

[14]

N. Cîndea and A. Münch, A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations, Calcolo, 52 (2015), 245-288. doi: 10.1007/s10092-014-0116-x.

[15]

J. -M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007.

[16]

R. Curtain and K. Morris, Transfer functions of distributed parameter systems: A tutorial, Automatica J. IFAC, 45 (2009), 1101-1116. doi: 10.1016/j.automatica.2009.01.008.

[17]

Y.-H. Dai and H. Zhang, Adaptive two-point stepsize gradient algorithm, Numer. Algorithms, 27 (2001), 377-385. doi: 10.1023/A:1013844413130.

[18]

A. Ern and J. -L. Guermond, Theory and Practice of Finite Elements, vol. 159 of Applied Mathematical Sciences, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[19]

F. Flandoli, Invertibility of Riccati operators and controllability of related systems, Systems Control Lett., 9 (1987), 65-72. doi: 10.1016/0167-6911(87)90010-7.

[20]

G. GrimmM. J. MessinaS. E. Tuna and A. R. Teel, Model predictive control: For want of a local control Lyapunov function, all is not lost, IEEE Trans. Automat. Control, 50 (2005), 546-558. doi: 10.1109/TAC.2005.847055.

[21]

L. Grüne, Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems, SIAM J. Control Optim., 48 (2009), 1206-1228. doi: 10.1137/070707853.

[22]

L. Grüne and A. Rantzer, On the infinite horizon performance of receding horizon controllers, IEEE Trans. Automat. Control, 53 (2008), 2100-2111. doi: 10.1109/TAC.2008.927799.

[23]

M. Gugat, Exponential stabilization of the wave equation by Dirichlet integral feedback, SIAM J. Control Optim., 53 (2015), 526-546. doi: 10.1137/140977023.

[24]

M. GugatG. Leugering and G. Sklyar, $L^p$-optimal boundary control for the wave equation, SIAM J. Control Optim., 44 (2005), 49-74. doi: 10.1137/S0363012903419212.

[25]

M. GugatE. Trélat and E. Zuazua, Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property, Systems Control Lett., 90 (2016), 61-70. doi: 10.1016/j.sysconle.2016.02.001.

[26]

W. Guo and B.-Z. Guo, Adaptive output feedback stabilization for one-dimensional wave equation with corrupted observation by harmonic disturbance, SIAM J. Control Optim., 51 (2013), 1679-1706. doi: 10.1137/120873212.

[27]

E. Hendrickson and I. Lasiecka, Numerical approximations of solutions to Riccati equations arising in boundary control problems for the wave equation, in Optimal control of differential equations (Athens, OH, 1993), vol. 160 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1994,111-132.

[28]

K. Ito and K. Kunisch, Receding horizon optimal control for infinite dimensional systems, ESAIM Control Optim. Calc. Var., 8 (2002), 741-760 (electronic). A tribute to J. L. Lions. doi: 10.1051/cocv:2002032.

[29]

A. Jadbabaie and J. Hauser, On the stability of receding horizon control with a general terminal cost, IEEE Trans. Automat. Control, 50 (2005), 674-678. doi: 10.1109/TAC.2005.846597.

[30]

C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 107 (1993), 117-129. doi: 10.1016/0045-7825(93)90170-3.

[31]

O. Karakashian and C. Makridakis, Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations, Math. Comp., 74 (2005), 85-102. doi: 10.1090/S0025-5718-04-01654-0.

[32]

A. KrönerK. Kunisch and B. Vexler, Semismooth Newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49 (2011), 830-858. doi: 10.1137/090766541.

[33]

K. KunischP. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244. doi: 10.1137/141001366.

[34]

J. E. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256. doi: 10.1137/0326068.

[35]

I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2(0, ∞; L_2(Γ))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390. doi: 10.1016/0022-0396(87)90025-8.

[36]

——, Sharp regularity theory for second order hyperbolic equations of Neumann type. I. L2 nonhomogeneous data, Ann. Mat. Pura Appl. (4), 157 (1990), 285-367. doi: 10.1007/BF01765322.

[37]

——, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, vol. 164 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0006880.

[38]

——, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224. doi: 10.1007/BF01182480.

[39]

——, Algebraic Riccati equations arising from systems with unbounded input-solution operator: applications to boundary control problems for wave and plate equations, Nonlinear Anal., 20 (1993), 659-695. doi: 10.1016/0362-546X(93)90026-O.

[40]

J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.

[41]

——, Contrôle des Systémes Distribués Singuliers, vol. 13 of Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Gauthier-Villars, Montrouge, 1983.

[42]

J. -L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.

[43]

——, Non-homogeneous Boundary Value Problems and Applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182.

[44]

D. Q. MayneJ. B. RawlingsC. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica J. IFAC, 36 (2000), 789-814. doi: 10.1016/S0005-1098(99)00214-9.

[45]

B. S. Mordukhovich and J. -P. Raymond, Neumann boundary control of hyperbolic equations with pointwise state constraints, SIAM J. Control Optim., 43 (2004/05), 1354-1372 (electronic). doi: 10.1137/S0363012903431177.

[46]

B. S. Mordukhovich and J.-P. Raymond, Dirichlet boundary control of hyperbolic equations in the presence of state constraints, Appl. Math. Optim., 49 (2004), 145-157. doi: 10.1007/BF02638149.

[47]

A. Münch and A. F. Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation, ESAIM Control Optim. Calc. Var., 13 (2007), 265-293 (electronic). doi: 10.1051/cocv:2007009.

[48]

M. Reble and F. Allgöwer, Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems, Automatica J. IFAC, 48 (2012), 1812-1817. doi: 10.1016/j.automatica.2012.05.067.

[49]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[50]

L. T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite difference space discretization of the $1-d$ wave equation, Adv. Comput. Math., 26 (2007), 337-365. doi: 10.1007/s10444-004-7629-9.

[51]

R. Triggiani, Exact boundary controllability on $L_2(Ω)× H^{-1}(Ω)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partialΩ$, and related problems, Appl. Math. Optim., 18 (1988), 241-277. doi: 10.1007/BF01443625.

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
Table4 
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.
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.
Table5 
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 $i = 0, \dots, r-1$.
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 $i = 0, \dots, r-1$.
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$
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$
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$
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$
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$
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$
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