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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

Birgit Jacob and  Hafida Laasri

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.

Keywords: infinite-dimensional non-autonomous control system, evolution family, port-Hamiltonian system, well-posedness.
Mathematics Subject Classification: 93C25, 47D06, 93C20.
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. 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.  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. 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.  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. 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.  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 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. 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.  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 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.  Google Scholar

[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.  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. 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.  Google Scholar

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