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Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids
Well-posedness of infinite-dimensional non-autonomous passive boundary control systems
University of Wuppertal, Work group Functional Analysis, 42097 Wuppertal, Germany |
We study a class of non-autonomous linear boundary control and observation systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.
References:
[1] |
P. Acquistapace and B. Terreni,
Classical solutions of nonautonomous Riccati equations arising in parabolic boundary control problems, Appl Math Optim, 39 (1999), 361-409.
doi: 10.1007/s002459900111. |
[2] |
B. Augner, Stability of Infnite-Dimensional Port-Hamiltonian System Via Dissipative Boundary Feedback, PhD thesis, University of Wuppertal, 2016. Google Scholar |
[3] |
B. Augner and B. Jacob,
Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems, Evolution Equations and Control Theory, 3 (2014), 207-229.
doi: 10.3934/eect.2014.3.207. |
[4] |
B. Augner, B. Jacob and H. Laasri,
On the right multiplicative perturbation of nonautonomous $L^p$-maximal regularity, J. Operator Theory, 74 (2015), 391-415.
doi: 10.7900/jot.2014jul31.2064. |
[5] |
C. Beattie, V. Mehrmann, H. Xu and H. Zwart, Linear port-hamiltonian descriptor systems, Math. Control Signals Systems, 30 (2018), Art. 17, 27 pp.
doi: 10.1007/s00498-018-0223-3. |
[6] |
H. Bounit and A. Idrissi,
Time-varying regular bilinear systems, SIAM J. Control and Optim, 47 (2008), 1097-1126.
doi: 10.1137/050632245. |
[7] |
J. H. Chen and G. Weiss,
Time-varying additive perturbations of well-posed linear systems, Math. Control Signals Systems, 27 (2015), 149-185.
doi: 10.1007/s00498-014-0136-8. |
[8] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1978. |
[9] |
R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[10] |
K. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[11] |
K. J. Engel and M. Bombieri,
A semigroup characterization of well-posed linear control systems, Semigroup Forum, 88 (2014), 366-396.
doi: 10.1007/s00233-013-9545-0. |
[12] |
H. O. Fattorini,
Boundary control systems, SIAM J. Control, (6) (1968), 349-385.
doi: 10.1137/0306025. |
[13] |
B. H. Haak, D. T. Hoang and E. M. Ouhabaz, Controllability and observability for non-autonomous evolution equations: The averaged Hautus test, Systems Control Lett, 133 (2019), 104524.
doi: 10.1016/j.sysconle.2019.104524. |
[14] |
B. Jacob, Time-Varying Infinite Dimensional State-Space Systems, PhD thesis, Bremen, 1995. Google Scholar |
[15] |
B. Jacob and J. Kaiser,
Well-posedness of systems of 1-D hyperbolic partial differential equations, J. Evol. Equ., 19 (2019), 91-109.
doi: 10.1007/s00028-018-0470-2. |
[16] |
B. Jacob, K. Morris and H. Zwart,
$C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.
doi: 10.1007/s00028-014-0271-1. |
[17] |
B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Operator Theory: Advances and Applications, 223. Linear Operators and Linear Systems. Birkhäuser/Springer Basel AG, Basel, 2012.
doi: 10.1007/978-3-0348-0399-1. |
[18] |
S. Hadd,
An evolution equation approach to non-autonomous linear systems with state, input and output delays, SIAM J. Control Optim, 45 (2006), 246-272.
doi: 10.1137/040612178. |
[19] |
S. Hadd, A. Rhandi and R. Schnaubelt,
Feedback theory for non-autonomous linear systems with input delays, IMA J. Math. Control Inform., 25 (2008), 85-110.
doi: 10.1093/imamci/dnm011. |
[20] |
T. Kato,
Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241-258.
|
[21] |
J. Kisynski,
Sur les operateurs de green des problemes de Cauchy abstraits, Studia Mathematica, 23 (1964), 285-328.
doi: 10.4064/sm-23-3-285-328. |
[22] |
M. Kurula, Well-posedness of time-varying linear systems, IEEE Transactions on Automatic Control (Early Access), 2019, 1–1, available from: https://arXiv.org/abs/1904.12367.
doi: 10.1109/TAC.2019.2954794. |
[23] |
Y. Le Gorrec, H. Zwart and B. Maschke,
Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim., 44 (2005), 1864-1892.
doi: 10.1137/040611677. |
[24] |
J. Malinen, O. Staffans and G. Weiss,
When is a linear system conservative?, Quart. Appl. Math., 64 (2006), 61-91.
doi: 10.1090/S0033-569X-06-00994-7. |
[25] |
G. Nickel,
On evolution semigroups and wellposedness of nonautonomous cauchy problems, Diss. Summ. Math., 1 (1996), 195-202.
|
[26] |
L. Paunonen and S. Pohjolainen,
Periodic output regulation for distributed parameter systems, Math Control Signals Syst, 24 (2012), 403-441.
doi: 10.1007/s00498-012-0087-x. |
[27] |
L. Paunonen,
Robust output regulation for continuous-time periodic systems, IEEE Trans. Automat. Control, 62 (2017), 4363-4375.
doi: 10.1109/TAC.2017.2654968. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
R. Schnaubelt, Well-Posedness and asymptotic behaviour of nonautonomous evolution equation, Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2002), 311–338. |
[30] |
R. Schnaubelt,
Feedbacks for nonautonomous regular linear systems, SIAM J. Control Optim., 41 (2002), 1141-1165.
doi: 10.1137/S036301290139169X. |
[31] |
R. Schnaubelt and G. Weiss,
Two classes of passive time-varying well-posed linear systems, Math. Control Signals Systems, 21 (2010), 265-301.
doi: 10.1007/s00498-010-0049-0. |
[32] |
O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197.![]() ![]() |
[33] |
O. Staffans and G. Weiss,
Transfer functions of regular linear systems. II. the system operator and the lax-phillips semigroup, Trans. Amer. Math. Soc., 354 (2002), 3229-3262.
doi: 10.1090/S0002-9947-02-02976-8. |
[34] |
H. Tanabe, Equation of Evolution, Pitman, London, 1979. |
[35] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[36] |
M. Tucsnak and G. Weiss,
Well-posed systems-the LTI case and beyond, Automatica J. IFAC, 50 (2014), 1757-1779.
doi: 10.1016/j.automatica.2014.04.016. |
[37] |
J. A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D thesis, Universiteit Twente, 2007, Available from: http://doc.utwente.nl/57842/1/thesis_Villegas.pdf. Google Scholar |
[38] |
J. A. Villegas, H. Zwart, Y. Le Gorrec and A. van der Schaft, Boundary control systems and the system node, IFAC World Congress, 38 2005,308–313.
doi: 10.3182/20050703-6-CZ-1902.00622. |
[39] |
G. Weiss,
Transfer functions of regular linear systems, part I: Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.
doi: 10.2307/2154655. |
[40] |
G. Weiss,
Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.
doi: 10.1007/BF02788172. |
[41] |
G. Weiss, The representation of regular linear systems on Hrt spaces, Control and Estimation of Distributed Parameter Systems (Vorau, 1988), Internat. Ser. Numer. Math., Birkhäuser, Basel, 91 (1989), 401–416. |
[42] |
H. Zwart, Y. Le Gorrec, B. Maschke and J. A. Villegas,
Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM Contr. Optim. Calc. Var., 16 (2010), 1077-1093.
doi: 10.1051/cocv/2009036. |
show all references
References:
[1] |
P. Acquistapace and B. Terreni,
Classical solutions of nonautonomous Riccati equations arising in parabolic boundary control problems, Appl Math Optim, 39 (1999), 361-409.
doi: 10.1007/s002459900111. |
[2] |
B. Augner, Stability of Infnite-Dimensional Port-Hamiltonian System Via Dissipative Boundary Feedback, PhD thesis, University of Wuppertal, 2016. Google Scholar |
[3] |
B. Augner and B. Jacob,
Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems, Evolution Equations and Control Theory, 3 (2014), 207-229.
doi: 10.3934/eect.2014.3.207. |
[4] |
B. Augner, B. Jacob and H. Laasri,
On the right multiplicative perturbation of nonautonomous $L^p$-maximal regularity, J. Operator Theory, 74 (2015), 391-415.
doi: 10.7900/jot.2014jul31.2064. |
[5] |
C. Beattie, V. Mehrmann, H. Xu and H. Zwart, Linear port-hamiltonian descriptor systems, Math. Control Signals Systems, 30 (2018), Art. 17, 27 pp.
doi: 10.1007/s00498-018-0223-3. |
[6] |
H. Bounit and A. Idrissi,
Time-varying regular bilinear systems, SIAM J. Control and Optim, 47 (2008), 1097-1126.
doi: 10.1137/050632245. |
[7] |
J. H. Chen and G. Weiss,
Time-varying additive perturbations of well-posed linear systems, Math. Control Signals Systems, 27 (2015), 149-185.
doi: 10.1007/s00498-014-0136-8. |
[8] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1978. |
[9] |
R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[10] |
K. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[11] |
K. J. Engel and M. Bombieri,
A semigroup characterization of well-posed linear control systems, Semigroup Forum, 88 (2014), 366-396.
doi: 10.1007/s00233-013-9545-0. |
[12] |
H. O. Fattorini,
Boundary control systems, SIAM J. Control, (6) (1968), 349-385.
doi: 10.1137/0306025. |
[13] |
B. H. Haak, D. T. Hoang and E. M. Ouhabaz, Controllability and observability for non-autonomous evolution equations: The averaged Hautus test, Systems Control Lett, 133 (2019), 104524.
doi: 10.1016/j.sysconle.2019.104524. |
[14] |
B. Jacob, Time-Varying Infinite Dimensional State-Space Systems, PhD thesis, Bremen, 1995. Google Scholar |
[15] |
B. Jacob and J. Kaiser,
Well-posedness of systems of 1-D hyperbolic partial differential equations, J. Evol. Equ., 19 (2019), 91-109.
doi: 10.1007/s00028-018-0470-2. |
[16] |
B. Jacob, K. Morris and H. Zwart,
$C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.
doi: 10.1007/s00028-014-0271-1. |
[17] |
B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Operator Theory: Advances and Applications, 223. Linear Operators and Linear Systems. Birkhäuser/Springer Basel AG, Basel, 2012.
doi: 10.1007/978-3-0348-0399-1. |
[18] |
S. Hadd,
An evolution equation approach to non-autonomous linear systems with state, input and output delays, SIAM J. Control Optim, 45 (2006), 246-272.
doi: 10.1137/040612178. |
[19] |
S. Hadd, A. Rhandi and R. Schnaubelt,
Feedback theory for non-autonomous linear systems with input delays, IMA J. Math. Control Inform., 25 (2008), 85-110.
doi: 10.1093/imamci/dnm011. |
[20] |
T. Kato,
Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241-258.
|
[21] |
J. Kisynski,
Sur les operateurs de green des problemes de Cauchy abstraits, Studia Mathematica, 23 (1964), 285-328.
doi: 10.4064/sm-23-3-285-328. |
[22] |
M. Kurula, Well-posedness of time-varying linear systems, IEEE Transactions on Automatic Control (Early Access), 2019, 1–1, available from: https://arXiv.org/abs/1904.12367.
doi: 10.1109/TAC.2019.2954794. |
[23] |
Y. Le Gorrec, H. Zwart and B. Maschke,
Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim., 44 (2005), 1864-1892.
doi: 10.1137/040611677. |
[24] |
J. Malinen, O. Staffans and G. Weiss,
When is a linear system conservative?, Quart. Appl. Math., 64 (2006), 61-91.
doi: 10.1090/S0033-569X-06-00994-7. |
[25] |
G. Nickel,
On evolution semigroups and wellposedness of nonautonomous cauchy problems, Diss. Summ. Math., 1 (1996), 195-202.
|
[26] |
L. Paunonen and S. Pohjolainen,
Periodic output regulation for distributed parameter systems, Math Control Signals Syst, 24 (2012), 403-441.
doi: 10.1007/s00498-012-0087-x. |
[27] |
L. Paunonen,
Robust output regulation for continuous-time periodic systems, IEEE Trans. Automat. Control, 62 (2017), 4363-4375.
doi: 10.1109/TAC.2017.2654968. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
R. Schnaubelt, Well-Posedness and asymptotic behaviour of nonautonomous evolution equation, Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2002), 311–338. |
[30] |
R. Schnaubelt,
Feedbacks for nonautonomous regular linear systems, SIAM J. Control Optim., 41 (2002), 1141-1165.
doi: 10.1137/S036301290139169X. |
[31] |
R. Schnaubelt and G. Weiss,
Two classes of passive time-varying well-posed linear systems, Math. Control Signals Systems, 21 (2010), 265-301.
doi: 10.1007/s00498-010-0049-0. |
[32] |
O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197.![]() ![]() |
[33] |
O. Staffans and G. Weiss,
Transfer functions of regular linear systems. II. the system operator and the lax-phillips semigroup, Trans. Amer. Math. Soc., 354 (2002), 3229-3262.
doi: 10.1090/S0002-9947-02-02976-8. |
[34] |
H. Tanabe, Equation of Evolution, Pitman, London, 1979. |
[35] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[36] |
M. Tucsnak and G. Weiss,
Well-posed systems-the LTI case and beyond, Automatica J. IFAC, 50 (2014), 1757-1779.
doi: 10.1016/j.automatica.2014.04.016. |
[37] |
J. A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems, Ph.D thesis, Universiteit Twente, 2007, Available from: http://doc.utwente.nl/57842/1/thesis_Villegas.pdf. Google Scholar |
[38] |
J. A. Villegas, H. Zwart, Y. Le Gorrec and A. van der Schaft, Boundary control systems and the system node, IFAC World Congress, 38 2005,308–313.
doi: 10.3182/20050703-6-CZ-1902.00622. |
[39] |
G. Weiss,
Transfer functions of regular linear systems, part I: Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.
doi: 10.2307/2154655. |
[40] |
G. Weiss,
Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.
doi: 10.1007/BF02788172. |
[41] |
G. Weiss, The representation of regular linear systems on Hrt spaces, Control and Estimation of Distributed Parameter Systems (Vorau, 1988), Internat. Ser. Numer. Math., Birkhäuser, Basel, 91 (1989), 401–416. |
[42] |
H. Zwart, Y. Le Gorrec, B. Maschke and J. A. Villegas,
Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM Contr. Optim. Calc. Var., 16 (2010), 1077-1093.
doi: 10.1051/cocv/2009036. |
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