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December  2021, 41(12): 5789-5824. doi: 10.3934/dcds.2021096

Stabilization of nonautonomous parabolic equations by a single moving actuator

1. 

Johann Radon Institute for Computational and Applied Mathematics, ÖAW, Altenbergerstr. 69, 4040 Linz, Austria

2. 

Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, Heinrichstr. 36, 8010 Graz, Austria

* Corresponding author

Received  November 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

It is shown that an internal control based on a moving indicator function is able to stabilize the state of parabolic equations evolving in rectangular domains. For proving the stabilizability result, we start with a control obtained from an oblique projection feedback based on a finite number of static actuators, then we used the continuity of the state when the control varies in a relaxation metric to construct a switching control where at each given instant of time only one of the static actuators is active, finally we construct the moving control by traveling between the static actuators.

Numerical computations are performed by a concatenation procedure following a receding horizon control approach. They confirm the stabilizing performance of the moving control.

Citation: Behzad Azmi, Karl Kunisch, Sérgio S. Rodrigues. Stabilization of nonautonomous parabolic equations by a single moving actuator. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5789-5824. doi: 10.3934/dcds.2021096
References:
[1]

A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: Controllability by means of low modes forcing, J. Math. Fluid Mech., 7 (2005), 108-152.  doi: 10.1007/s00021-004-0110-1.

[2]

A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys., 265 (2006), 673-697.  doi: 10.1007/s00220-006-0002-8.

[3]

B. Azmi and K. Kunisch, A hybrid finite-dimensional RHC for stabilization of time-varying parabolic equations, SIAM J. Control Optim., 57 (2019), 3496-3526.  doi: 10.1137/19M1239787.

[4]

B. Azmi and K. Kunisch, Analysis of the Barzilai-Borwein step-sizes for problems in Hilbert spaces, J. Optim. Theory Appl., 185 (2020), 819-844.  doi: 10.1007/s10957-020-01677-y.

[5]

M. Badra, Feedback stabilization of the 2-D and 3-D Navier–Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968.  doi: 10.1051/cocv:2008059.

[6]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.

[7]

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.

[8]

T. BreitenK. Kunisch and S. S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh–Nagumo type, SIAM J. Control Optim., 55 (2017), 2684-2713.  doi: 10.1137/15M1038165.

[9]

C. CastroN. Cîndea and A. Münch, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim., 52 (2014), 4027-4056.  doi: 10.1137/140956129.

[10]

C. Castro and E. Zuazua, Unique continuation and control for the heat equation from an oscillating lower dimensional manifold, SIAM J. Control Optim., 43 (2005), 1400-1434.  doi: 10.1137/S0363012903430317.

[11]

F. W. Chaves-SilvaL. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control, J. Math. Pures Appl., 101 (2014), 198-222.  doi: 10.1016/j.matpur.2013.05.009.

[12]

Y.-H. Dai and R. Fletcher, New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program., 106 (2006), 403-421.  doi: 10.1007/s10107-005-0595-2.

[13]

M. A. Demetriou, Guidance of mobile actuator-plus-sensor networks for improved control and estimation of distributed parameter systems, IEEE Trans. Automat. Control, 55 (2010), 1570-1584.  doi: 10.1109/TAC.2010.2042229.

[14] R. V. Gamkrelidze, Principles of Optimal Control Theory, Plenum Press, 1978.  doi: 10.1007/978-1-4684-7398-8.
[15]

M. L. J. Hautus, Stabilization controllability and observability of linear autonomous systems, Indag. Math. (Proceedings), 73 (1970), 448-455.  doi: 10.1016/S1385-7258(70)80049-X.

[16]

Y. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 41 (2003), 1263-1285.  doi: 10.1137/S0036142901385659.

[17]

M. Hintermüller and K. Kunisch, Path-following methods for a class of constrained minimization problems in function space, SIAM J. Optim., 17 (2006), 159-187.  doi: 10.1137/040611598.

[18]

A. Khapalov, $L^\infty$-exact observability of the heat equation with scanning pointwise sensor, SIAM J. Control Optim., 32 (1994), 1037-1051.  doi: 10.1137/S036301299222737X.

[19]

A. Khapalov, Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls, ESAIM Control, Optim. Calc. Var., 4 (1999), 83-98.  doi: 10.1051/cocv:1999104.

[20]

A. Khapalov, Mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation, SIAM J. Control Optim., 40 (2001), 231-252.  doi: 10.1137/S0363012999358038.

[21]

A. Kröner and S. S. Rodrigues, Internal exponential stabilization to a nonstationary solution for 1D Burgers equations with piecewise constant controls, in Proceedings of the 2015 European Control Conference (ECC), Linz, Austria, 2015, 2676–2681. doi: 10.1109/ECC.2015.7330942.

[22]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., 25 (2019), 67. doi: 10.1051/cocv/2018054.

[23]

K. Kunisch and D. A. Souza, On the one-dimensional nonlinear monodomain equations with moving controls, J. Math. Pures Appl., 117 (2018), 94-122.  doi: 10.1016/j.matpur.2018.05.003.

[24]

P. MartinL. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control, SIAM J. Control Optim., 51 (2013), 660-684.  doi: 10.1137/110856150.

[25]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Syst., 30 (2018), 11. doi: 10.1007/s00498-018-0218-0.

[26]

S. S. Rodrigues, Navier–Stokes equation on the Rectangle: Controllability by means of low modes forcing, J. Dyn. Control Syst., 12 (2006), 517-562.  doi: 10.1007/s10883-006-0004-z.

[27]

S. S. Rodrigues, Methods of Geometric Control Theory in Problems of Mathematical Physics, Ph.D Thesis, Universidade de Aveiro, Portugal, 2008.

[28]

S. S. Rodrigues, Feedback boundary stabilization to trajectories for 3D Navier–Stokes equations, Appl. Math. Optim.. doi: 10.1007/s00245-017-9474-5.

[29]

S. S. Rodrigues, Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems, Evol. Equ. Control Theory, 9 (2020), 635-672.  doi: 10.3934/eect.2020027.

[30]

S. S. Rodrigues, Oblique projection output-based feedback stabilization of nonautonomous parabolic equations, Automatica J. IFAC, 129 (2021), 109621. doi: 10.1016/j.automatica.2021.109621.

[31]

S. S. Rodrigues and K. Sturm, On the explicit feedback stabilization of one-dimensional linear nonautonomous parabolic equations via oblique projections, IMA J. Math. Control Inform., 37 (2020), 175-207.  doi: 10.1093/imamci/dny045.

[32]

A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 521-537.  doi: 10.1016/j.anihpc.2006.04.002.

[33]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Basel, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[34]

J. Zabczyk, Mathematical Control Theory: An Introduction, Systems Control Found. Appl., Birkhäuser, Boston, 1992. doi: 10.1007/978-0-8176-4733-9.

show all references

References:
[1]

A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: Controllability by means of low modes forcing, J. Math. Fluid Mech., 7 (2005), 108-152.  doi: 10.1007/s00021-004-0110-1.

[2]

A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys., 265 (2006), 673-697.  doi: 10.1007/s00220-006-0002-8.

[3]

B. Azmi and K. Kunisch, A hybrid finite-dimensional RHC for stabilization of time-varying parabolic equations, SIAM J. Control Optim., 57 (2019), 3496-3526.  doi: 10.1137/19M1239787.

[4]

B. Azmi and K. Kunisch, Analysis of the Barzilai-Borwein step-sizes for problems in Hilbert spaces, J. Optim. Theory Appl., 185 (2020), 819-844.  doi: 10.1007/s10957-020-01677-y.

[5]

M. Badra, Feedback stabilization of the 2-D and 3-D Navier–Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968.  doi: 10.1051/cocv:2008059.

[6]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.

[7]

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.

[8]

T. BreitenK. Kunisch and S. S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh–Nagumo type, SIAM J. Control Optim., 55 (2017), 2684-2713.  doi: 10.1137/15M1038165.

[9]

C. CastroN. Cîndea and A. Münch, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim., 52 (2014), 4027-4056.  doi: 10.1137/140956129.

[10]

C. Castro and E. Zuazua, Unique continuation and control for the heat equation from an oscillating lower dimensional manifold, SIAM J. Control Optim., 43 (2005), 1400-1434.  doi: 10.1137/S0363012903430317.

[11]

F. W. Chaves-SilvaL. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control, J. Math. Pures Appl., 101 (2014), 198-222.  doi: 10.1016/j.matpur.2013.05.009.

[12]

Y.-H. Dai and R. Fletcher, New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program., 106 (2006), 403-421.  doi: 10.1007/s10107-005-0595-2.

[13]

M. A. Demetriou, Guidance of mobile actuator-plus-sensor networks for improved control and estimation of distributed parameter systems, IEEE Trans. Automat. Control, 55 (2010), 1570-1584.  doi: 10.1109/TAC.2010.2042229.

[14] R. V. Gamkrelidze, Principles of Optimal Control Theory, Plenum Press, 1978.  doi: 10.1007/978-1-4684-7398-8.
[15]

M. L. J. Hautus, Stabilization controllability and observability of linear autonomous systems, Indag. Math. (Proceedings), 73 (1970), 448-455.  doi: 10.1016/S1385-7258(70)80049-X.

[16]

Y. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 41 (2003), 1263-1285.  doi: 10.1137/S0036142901385659.

[17]

M. Hintermüller and K. Kunisch, Path-following methods for a class of constrained minimization problems in function space, SIAM J. Optim., 17 (2006), 159-187.  doi: 10.1137/040611598.

[18]

A. Khapalov, $L^\infty$-exact observability of the heat equation with scanning pointwise sensor, SIAM J. Control Optim., 32 (1994), 1037-1051.  doi: 10.1137/S036301299222737X.

[19]

A. Khapalov, Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls, ESAIM Control, Optim. Calc. Var., 4 (1999), 83-98.  doi: 10.1051/cocv:1999104.

[20]

A. Khapalov, Mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation, SIAM J. Control Optim., 40 (2001), 231-252.  doi: 10.1137/S0363012999358038.

[21]

A. Kröner and S. S. Rodrigues, Internal exponential stabilization to a nonstationary solution for 1D Burgers equations with piecewise constant controls, in Proceedings of the 2015 European Control Conference (ECC), Linz, Austria, 2015, 2676–2681. doi: 10.1109/ECC.2015.7330942.

[22]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., 25 (2019), 67. doi: 10.1051/cocv/2018054.

[23]

K. Kunisch and D. A. Souza, On the one-dimensional nonlinear monodomain equations with moving controls, J. Math. Pures Appl., 117 (2018), 94-122.  doi: 10.1016/j.matpur.2018.05.003.

[24]

P. MartinL. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control, SIAM J. Control Optim., 51 (2013), 660-684.  doi: 10.1137/110856150.

[25]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Syst., 30 (2018), 11. doi: 10.1007/s00498-018-0218-0.

[26]

S. S. Rodrigues, Navier–Stokes equation on the Rectangle: Controllability by means of low modes forcing, J. Dyn. Control Syst., 12 (2006), 517-562.  doi: 10.1007/s10883-006-0004-z.

[27]

S. S. Rodrigues, Methods of Geometric Control Theory in Problems of Mathematical Physics, Ph.D Thesis, Universidade de Aveiro, Portugal, 2008.

[28]

S. S. Rodrigues, Feedback boundary stabilization to trajectories for 3D Navier–Stokes equations, Appl. Math. Optim.. doi: 10.1007/s00245-017-9474-5.

[29]

S. S. Rodrigues, Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems, Evol. Equ. Control Theory, 9 (2020), 635-672.  doi: 10.3934/eect.2020027.

[30]

S. S. Rodrigues, Oblique projection output-based feedback stabilization of nonautonomous parabolic equations, Automatica J. IFAC, 129 (2021), 109621. doi: 10.1016/j.automatica.2021.109621.

[31]

S. S. Rodrigues and K. Sturm, On the explicit feedback stabilization of one-dimensional linear nonautonomous parabolic equations via oblique projections, IMA J. Math. Control Inform., 37 (2020), 175-207.  doi: 10.1093/imamci/dny045.

[32]

A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 521-537.  doi: 10.1016/j.anihpc.2006.04.002.

[33]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Basel, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[34]

J. Zabczyk, Mathematical Control Theory: An Introduction, Systems Control Found. Appl., Birkhäuser, Boston, 1992. doi: 10.1007/978-0-8176-4733-9.

Figure 1.  An internal moving actuator with support $ \overline{\omega(c(t))}\subset\overline{\Omega} $
Figure 2.  Supports of the static actuators. Case $ \Omega\subset {\mathbb R}^2 $
Figure 3.  Example 5.1: Numerical results for $ \beta = 0.1 $
Figure 4.  Example 5.1: Numerical results for $ \beta = 0.5 $
Figure 5.  Example 5.2
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