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June  2021, 10(2): 385-409. doi: 10.3934/eect.2020072

Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

University of Wuppertal, Work group Functional Analysis, 42097 Wuppertal, Germany

The second author gratefully acknowledges support from Deutsche Forschungsgemeinschaft (Grant LA 4197/1-1)

Received  June 2019 Revised  April 2020 Published  June 2020

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.

Citation: Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations & Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072
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.  Google Scholar

[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.  Google Scholar

[4]

B. AugnerB. 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.  Google Scholar

[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.  Google Scholar

[6]

H. Bounit and A. Idrissi, Time-varying regular bilinear systems, SIAM J. Control and Optim, 47 (2008), 1097-1126.  doi: 10.1137/050632245.  Google Scholar

[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.  Google Scholar

[8]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1978.  Google Scholar

[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.  Google Scholar

[10]

K. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

[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.  Google Scholar

[12]

H. O. Fattorini, Boundary control systems, SIAM J. Control, (6) (1968), 349-385.  doi: 10.1137/0306025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. JacobK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. HaddA. 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.  Google Scholar

[20]

T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241-258.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Y. Le GorrecH. 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.  Google Scholar

[24]

J. MalinenO. 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.  Google Scholar

[25]

G. Nickel, On evolution semigroups and wellposedness of nonautonomous cauchy problems, Diss. Summ. Math., 1 (1996), 195-202.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[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.  Google Scholar

[30]

R. Schnaubelt, Feedbacks for nonautonomous regular linear systems, SIAM J. Control Optim., 41 (2002), 1141-1165.  doi: 10.1137/S036301290139169X.  Google Scholar

[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.  Google Scholar

[32] O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar
[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.  Google Scholar

[34]

H. Tanabe, Equation of Evolution, Pitman, London, 1979.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.  Google Scholar

[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.  Google Scholar

[42]

H. ZwartY. Le GorrecB. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[4]

B. AugnerB. 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.  Google Scholar

[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.  Google Scholar

[6]

H. Bounit and A. Idrissi, Time-varying regular bilinear systems, SIAM J. Control and Optim, 47 (2008), 1097-1126.  doi: 10.1137/050632245.  Google Scholar

[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.  Google Scholar

[8]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1978.  Google Scholar

[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.  Google Scholar

[10]

K. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

[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.  Google Scholar

[12]

H. O. Fattorini, Boundary control systems, SIAM J. Control, (6) (1968), 349-385.  doi: 10.1137/0306025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. JacobK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. HaddA. 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.  Google Scholar

[20]

T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 17 (1970), 241-258.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Y. Le GorrecH. 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.  Google Scholar

[24]

J. MalinenO. 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.  Google Scholar

[25]

G. Nickel, On evolution semigroups and wellposedness of nonautonomous cauchy problems, Diss. Summ. Math., 1 (1996), 195-202.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[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.  Google Scholar

[30]

R. Schnaubelt, Feedbacks for nonautonomous regular linear systems, SIAM J. Control Optim., 41 (2002), 1141-1165.  doi: 10.1137/S036301290139169X.  Google Scholar

[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.  Google Scholar

[32] O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar
[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.  Google Scholar

[34]

H. Tanabe, Equation of Evolution, Pitman, London, 1979.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.  Google Scholar

[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.  Google Scholar

[42]

H. ZwartY. Le GorrecB. 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.  Google Scholar

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