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On the stability of boundary equilibria in Filippov systems
Boundary stabilization of non-diagonal systems by proportional feedback forms
Alexandru Ioan Cuza University, Department of Mathematics, and Octav Mayer Institute of Mathematics (Romanian Academy), Carol I No. 11, 700506 Iaşi, Romania |
In this work, we are concerned with the problem of boundary exponential stabilization, in a Hilbert space $ H $, of parabolic type equations, namely equations for which their linear parts generate analytic $ C_0- $semigroups. We consider the case where the projection of the linear leading operator, on a given Riesz basis of $ H $, is non-diagonal. We do not assume that the linear operator has compact resolvent. Therefore, the Riesz basis is not necessarily an eigenbasis. The boundary stabilizer is given in a simple linear feedback form, of finite-dimensional structure, involving only the Riesz basis. To illustrate the results, at the end of the paper, we provide an example of stabilization of a fourth-order evolution equation on the half-axis.
References:
[1] |
L. Arnold, H. Crauel and V. Wihstutz,
Stabilization of linear systems by noise, SIAM J. Control Optim., 21 (1983), 451-461.
doi: 10.1137/0321027. |
[2] |
M. Badra and T. Takahashi,
Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: application to the Navier-Stokes system, SIAM J Control Optim., 49 (2011), 420-463.
doi: 10.1137/090778146. |
[3] |
V. Barbu,
Boundary Stabilization of Equilibrium Solutions to Parabolic Equations, IEEE Trans. Autom. Control, 58 (2013), 2416-2420.
doi: 10.1109/TAC.2013.2254013. |
[4] |
V. Barbu, Controllability and Stabilization of Parabolic Equations, Birkhäuser-Springer, Basel, 2018.
doi: 10.1007/978-3-319-76666-9. |
[5] |
V. Barbu, I. Lasiecka and R. Triggiani,
Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high-and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746.
doi: 10.1016/j.na.2005.09.012. |
[6] |
M. G. Burgos and L. Teresa,
Some results on controllability for linear and nonlinear heat equations in unbounded domains, Adv. Differ. Equ., 12 (2007), 1201-1240.
|
[7] |
A. V. Filinovskii,
Stabilization of the solutions of the wave equation in unbounded domains, Mat. Sb., 187 (1996), 131-160.
doi: 10.1070/SM1996v187n06ABEH000141. |
[8] |
A. V. Gorshkov,
Stabilization of the one-dimensional heat equation on a semibounded rod, Russ. Math. Surv., 56 (2001), 409-410.
doi: 10.1070/RM2001v056n02ABEH000388. |
[9] |
M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs (Advances in Design and Control), SIAM, 2008.
doi: 10.1137/1.9780898718607. |
[10] |
I. Lasiecka and R. Triggiani,
Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlin. Anal. Theory Meth. Appl., 121 (2015), 424-446.
doi: 10.1016/j.na.2015.03.012. |
[11] |
I. Lasiecka and R. Triggiani,
Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations, SIAM J. Control Optim., 21 (1983), 766-803.
doi: 10.1137/0321047. |
[12] |
H. Lian, J. Zhao and R. P. Agarwal, Upper and lower solution method for nth-order BVPs on an infinite interval, Bound. Value Probl. (2014), 17 pp.
doi: 10.1186/1687-2770-2014-100. |
[13] |
A. Lunardi,
Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\mathbb{R}^n$, Studia Math., 128 (1998), 171-198.
doi: 10.4064/sm-128-2-171-198. |
[14] |
S. Micu and E. Zuazua,
On the lack of null-controllability of the heat equation on the half-line, Trans. Am. Math. Soc., 353 (2001), 1635-1659.
doi: 10.1090/S0002-9947-00-02665-9. |
[15] |
F. Kh. Mukminov and I. M. Bikkulov,
Stabilization of the norm of the solution of a mixed problem in an unbounded domain for parabolic equations of orders 4 and 6, Mat. Sb., 195 (2004), 115-142.
doi: 10.1070/SM2004v195n03ABEH000810. |
[16] |
I. Munteanu, Boundary Stabilization of Parabolic Equations, Birkhauser-Springer, Basel, 2019.
doi: 10.1007/978-3-030-11099-4. |
[17] |
I. Munteanu, Exponential stabilization of the semilinear heat equation with nonlocal boundary conditions, J. Math. Analysis Appl., 492 (2020), art. no. 124512.
doi: 10.1016/j.jmaa.2020.124512. |
[18] |
A. Preumont, Collocated versus non-collocated control. In: Vibration Control of Active Structures. Solid Mechanics and Its Applications, Springer, Dordrecht, 1997.
doi: 10.1007/978-94-011-5654-7. |
[19] |
D. L. Russell,
A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies Appl. Math., 52 (1973), 189-211.
doi: 10.1002/sapm1973523189. |
[20] |
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. |
[21] |
R. Triggiani,
Boundary feedback stabilization of parabolic equations, Appl. Math. Optim., 6 (1980), 201-220.
doi: 10.1007/BF01442895. |
show all references
References:
[1] |
L. Arnold, H. Crauel and V. Wihstutz,
Stabilization of linear systems by noise, SIAM J. Control Optim., 21 (1983), 451-461.
doi: 10.1137/0321027. |
[2] |
M. Badra and T. Takahashi,
Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: application to the Navier-Stokes system, SIAM J Control Optim., 49 (2011), 420-463.
doi: 10.1137/090778146. |
[3] |
V. Barbu,
Boundary Stabilization of Equilibrium Solutions to Parabolic Equations, IEEE Trans. Autom. Control, 58 (2013), 2416-2420.
doi: 10.1109/TAC.2013.2254013. |
[4] |
V. Barbu, Controllability and Stabilization of Parabolic Equations, Birkhäuser-Springer, Basel, 2018.
doi: 10.1007/978-3-319-76666-9. |
[5] |
V. Barbu, I. Lasiecka and R. Triggiani,
Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high-and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746.
doi: 10.1016/j.na.2005.09.012. |
[6] |
M. G. Burgos and L. Teresa,
Some results on controllability for linear and nonlinear heat equations in unbounded domains, Adv. Differ. Equ., 12 (2007), 1201-1240.
|
[7] |
A. V. Filinovskii,
Stabilization of the solutions of the wave equation in unbounded domains, Mat. Sb., 187 (1996), 131-160.
doi: 10.1070/SM1996v187n06ABEH000141. |
[8] |
A. V. Gorshkov,
Stabilization of the one-dimensional heat equation on a semibounded rod, Russ. Math. Surv., 56 (2001), 409-410.
doi: 10.1070/RM2001v056n02ABEH000388. |
[9] |
M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs (Advances in Design and Control), SIAM, 2008.
doi: 10.1137/1.9780898718607. |
[10] |
I. Lasiecka and R. Triggiani,
Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlin. Anal. Theory Meth. Appl., 121 (2015), 424-446.
doi: 10.1016/j.na.2015.03.012. |
[11] |
I. Lasiecka and R. Triggiani,
Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations, SIAM J. Control Optim., 21 (1983), 766-803.
doi: 10.1137/0321047. |
[12] |
H. Lian, J. Zhao and R. P. Agarwal, Upper and lower solution method for nth-order BVPs on an infinite interval, Bound. Value Probl. (2014), 17 pp.
doi: 10.1186/1687-2770-2014-100. |
[13] |
A. Lunardi,
Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\mathbb{R}^n$, Studia Math., 128 (1998), 171-198.
doi: 10.4064/sm-128-2-171-198. |
[14] |
S. Micu and E. Zuazua,
On the lack of null-controllability of the heat equation on the half-line, Trans. Am. Math. Soc., 353 (2001), 1635-1659.
doi: 10.1090/S0002-9947-00-02665-9. |
[15] |
F. Kh. Mukminov and I. M. Bikkulov,
Stabilization of the norm of the solution of a mixed problem in an unbounded domain for parabolic equations of orders 4 and 6, Mat. Sb., 195 (2004), 115-142.
doi: 10.1070/SM2004v195n03ABEH000810. |
[16] |
I. Munteanu, Boundary Stabilization of Parabolic Equations, Birkhauser-Springer, Basel, 2019.
doi: 10.1007/978-3-030-11099-4. |
[17] |
I. Munteanu, Exponential stabilization of the semilinear heat equation with nonlocal boundary conditions, J. Math. Analysis Appl., 492 (2020), art. no. 124512.
doi: 10.1016/j.jmaa.2020.124512. |
[18] |
A. Preumont, Collocated versus non-collocated control. In: Vibration Control of Active Structures. Solid Mechanics and Its Applications, Springer, Dordrecht, 1997.
doi: 10.1007/978-94-011-5654-7. |
[19] |
D. L. Russell,
A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies Appl. Math., 52 (1973), 189-211.
doi: 10.1002/sapm1973523189. |
[20] |
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. |
[21] |
R. Triggiani,
Boundary feedback stabilization of parabolic equations, Appl. Math. Optim., 6 (1980), 201-220.
doi: 10.1007/BF01442895. |
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