doi: 10.3934/eect.2022017
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Well-posedness and stability of non-autonomous semilinear input-output systems

Fraunhofer Institute for Industrial Mathematics (ITWM), Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany

Received  January 2021 Revised  January 2022 Early access April 2022

We establish well-posedness results for non-autonomous semilinear input-output systems, the central assumption being the scattering-passivity of the considered semilinear system. Along the way, we also establish global stability estimates. We consider both systems with distributed control and observation and systems with boundary control and observation, and we treat them in a unified manner. Applications are given to nonlinearly controlled collocated systems and to nonlinearly controlled port-Hamiltonian systems.

Citation: Jochen Schmid. Well-posedness and stability of non-autonomous semilinear input-output systems. Evolution Equations and Control Theory, doi: 10.3934/eect.2022017
References:
[1]

B. Augner, Stabilisation of infinite-dimensional port-Hamiltonian systems via dissipative boundary feedback, PhD thesis, Universität Wuppertal, 2016.

[2]

B. Augner and H. Laasri, Exponential stability for infinite-dimensional non-autonomous port-Hamiltonian systems, Syst. Contr. Lett., 144 (2020), 104757, 11 pp. doi: 10.1016/j.sysconle.2020.104757.

[3]

S. BouliteA. Idrissi and L. Maniar, Controllability of semilinear boundary problems with nonlocal initial conditions, J. Math. Anal. Appl., 316 (2006), 566-578.  doi: 10.1016/j.jmaa.2005.05.006.

[4]

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

[5]

J.-H. Chen and G. Weiss, Time-varying additive perturbations of well-posed linear systems, Math. Contr. Signals Systems, 27 (2015), 149-185.  doi: 10.1007/s00498-014-0136-8.

[6]

F. H. ClarkeYu. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.

[7]

R. Curtain and H. Zwart, Stabilization of collocated systems by nonlinear boundary control, Syst. Contr. Lett., 96 (2016), 11-14.  doi: 10.1016/j.sysconle.2016.06.014.

[8]

S. DashkovskiyO. Kapustyan and J. Schmid, A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations, Math. Contr. Sign. Syst., 32 (2020), 309-326.  doi: 10.1007/s00498-020-00256-w.

[9]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

[10]

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

[11]

B. Haak, D.-T. Hoang and E.-M. Ouhabaz, Controllability and observability for non-autonomous evolution equations: The averaged Hautus test, Syst. Contr. Lett., 133 (2019), 104524, 8 pp. doi: 10.1016/j.sysconle.2019.104524.

[12]

S. Hadd, An evolution equation approach to non-autonomous linear systems with state, input and output delays, SIAM J. Contr. Optim., 45 (2006), 246-272.  doi: 10.1137/040612178.

[13]

A. HastirF. Califano and H. Zwart, Well-posedness of infinite-dimensional linear systems with nonlinear feedback, Syst. Contr. Lett., 128 (2019), 19-25.  doi: 10.1016/j.sysconle.2019.04.002.

[14]

B. JacobV. Dragan and A. J. Pritchard, Infinite-dimensional time-varying systems with nonlinear output feedback, Integral Equations Operator Theory, 22 (1995), 440-462.  doi: 10.1007/BF01203385.

[15]

B. Jacob and H. Laasri, Well-posedness of infinite-dimensional non-autonomous passive boundary control systems, Evol. Equ. Control Theory, 10 (2021), 385-409.  doi: 10.3934/eect.2020072.

[16]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Birkhäuser, 2012. doi: 10.1007/978-3-0348-0399-1.

[17]

T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan, 5 (1953), 208-234.  doi: 10.2969/jmsj/00520208.

[18]

T. Kato, On linear differential equations in Banach spaces, Comm. Pure Appl. Math., 9 (1956), 479-486.  doi: 10.1002/cpa.3160090319.

[19]

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

[20]

J. Kisyński, Sur les opérateurs de Green des problèmes de Cauchy abstraits, Stud. Math., 23 (1963/64), 285-328.  doi: 10.4064/sm-23-3-285-328.

[21]

M. MiletićD. StürzerA. Arnold and A. Kugi, Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system, IEEE Trans. Autom. Contr., 61 (2016), 2782-2795.  doi: 10.1109/TAC.2015.2499604.

[22]

A. Mironchenko and C. Prieur, Input-to-state stability of infinite-dimensional systems: recent results and open questions, SIAM Rev., 62 (2020), 529-614.  doi: 10.1137/19M1291248.

[23]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Contr., 63 (2018), 1602-1617.  doi: 10.1109/tac.2017.2756341.

[24]

G. Nickel, Evolution semigroups and product formulas for nonautonomous Cauchy problems, Math Nachr., 212 (2000), 101-116.  doi: 10.1002/(SICI)1522-2616(200004)212:1<101::AID-MANA101>3.0.CO;2-3.

[25]

G. Nickel and R. Schnaubelt, An extension of Kato's stability condition for non-autonomous Cauchy problems, Taiwanese J. Math., 2 (1998), 483-496.  doi: 10.11650/twjm/1500407019.

[26]

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

[27]

J. Prüß, On semilinear evolution equations in Banach spaces, J. Reine Angew. Math., 303 (1978), 144-158.  doi: 10.1515/crll.1978.303-304.144.

[28]

J. Prüß, A note on strict solutions to semilinear evolution equations, Math. Z., 171 (1980), 285-288.  doi: 10.1007/BF01214993.

[29]

W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, 1991.

[30]

J. Schmid, Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar, J. Evol. Equ., 16 (2016), 21-50.  doi: 10.1007/s00028-015-0291-5.

[31]

J. Schmid, Adiabatic Theorems for General Linear Operators and Well-Posedness of Linear Evolution Equations, PhD thesis, Universität Stuttgart, 2015. doi: 10.18419/opus-5178.

[32]

J. Schmid, Weak input-to-state stability: Characterizations and counterexamples, Math. Contr. Sign. Syst., 31 (2019), 433-454.  doi: 10.1007/s00498-019-00248-5.

[33]

J. Schmid, Stabilization of port-Hamiltonian systems with discontinuous energy densities, Evol. Equ. Contr. Th., (2022). doi: 10.3934/eect.2021063.

[34]

J. Schmid, S. Dashkovskiy, B. Jacob and H. Laasri, Well-posedness of non-autonomous semilinear systems, Conference Proceedings of the 11th Symposium on Nonlinear Control Systems, IFAC-PapersOnLine, 52 (2019), 216–220. doi: 10.1016/j.ifacol.2019.11.781.

[35]

J. Schmid and M. Griesemer, Kato's theorem on the integration of non-autonomous evolution equations, Math. Phys. Anal. Geom., 17 (2014), 265-271.  doi: 10.1007/s11040-014-9154-5.

[36]

J. Schmid, O. Kapustyan and S. Dashkovskiy, Asymptotic gain results for attractors of semilinear systems, Math. Contr. Rel. Fields, (2021). doi: 10.3934/mcrf.2021044.

[37]

J. Schmid and M. Griesemer, Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces, Math. Nachr., 290 (2017), 435-441.  doi: 10.1002/mana.201500052.

[38]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances., ESAIM Contr. Optim. Calc. Var., 27 (2021), Paper No. 53, 37 pp. doi: 10.1051/cocv/2021051.

[39]

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

[40]

R. Schnaubelt and G. Weiss, Two classes of passive time-varying well-posed linear systems, Math. Contr. Sign. Syst., 21 (2010), 265-301.  doi: 10.1007/s00498-010-0049-0.

[41]

F. L. Schwenninger, Input-to-state stability for parabolic boundary control: Linear and semi-linear systems, Control Theory of Infinite-Dimensional Systems, Operator Theory: Advances and Applications, 277 (2020), 83-116.  doi: 10.1007/978-3-030-35898-3_4.

[42]

N. Skrepek, Well-posedness of linear first order port-Hamiltonian systems on multidimensional spatial domains, Evol. Equ. Control Theory, 10 (2021), 965-1006.  doi: 10.3934/eect.2020098.

[43]

M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Contr. Sign. Syst., 2 (1989), 265-285.  doi: 10.1007/BF02551387.

[44]

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

[45]

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.

[46]

A. van der Schaft, $L^2$-Gain and Passivity Techniques in Nonlinear Control, 3rd edition, Springer, 2017. doi: 10.1007/978-3-319-49992-5.

[47]

J. C. Willems, Dissipative dynamical systems. Part I: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351.  doi: 10.1007/BF00276493.

show all references

References:
[1]

B. Augner, Stabilisation of infinite-dimensional port-Hamiltonian systems via dissipative boundary feedback, PhD thesis, Universität Wuppertal, 2016.

[2]

B. Augner and H. Laasri, Exponential stability for infinite-dimensional non-autonomous port-Hamiltonian systems, Syst. Contr. Lett., 144 (2020), 104757, 11 pp. doi: 10.1016/j.sysconle.2020.104757.

[3]

S. BouliteA. Idrissi and L. Maniar, Controllability of semilinear boundary problems with nonlocal initial conditions, J. Math. Anal. Appl., 316 (2006), 566-578.  doi: 10.1016/j.jmaa.2005.05.006.

[4]

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

[5]

J.-H. Chen and G. Weiss, Time-varying additive perturbations of well-posed linear systems, Math. Contr. Signals Systems, 27 (2015), 149-185.  doi: 10.1007/s00498-014-0136-8.

[6]

F. H. ClarkeYu. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149 (1998), 69-114.  doi: 10.1006/jdeq.1998.3476.

[7]

R. Curtain and H. Zwart, Stabilization of collocated systems by nonlinear boundary control, Syst. Contr. Lett., 96 (2016), 11-14.  doi: 10.1016/j.sysconle.2016.06.014.

[8]

S. DashkovskiyO. Kapustyan and J. Schmid, A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations, Math. Contr. Sign. Syst., 32 (2020), 309-326.  doi: 10.1007/s00498-020-00256-w.

[9]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

[10]

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

[11]

B. Haak, D.-T. Hoang and E.-M. Ouhabaz, Controllability and observability for non-autonomous evolution equations: The averaged Hautus test, Syst. Contr. Lett., 133 (2019), 104524, 8 pp. doi: 10.1016/j.sysconle.2019.104524.

[12]

S. Hadd, An evolution equation approach to non-autonomous linear systems with state, input and output delays, SIAM J. Contr. Optim., 45 (2006), 246-272.  doi: 10.1137/040612178.

[13]

A. HastirF. Califano and H. Zwart, Well-posedness of infinite-dimensional linear systems with nonlinear feedback, Syst. Contr. Lett., 128 (2019), 19-25.  doi: 10.1016/j.sysconle.2019.04.002.

[14]

B. JacobV. Dragan and A. J. Pritchard, Infinite-dimensional time-varying systems with nonlinear output feedback, Integral Equations Operator Theory, 22 (1995), 440-462.  doi: 10.1007/BF01203385.

[15]

B. Jacob and H. Laasri, Well-posedness of infinite-dimensional non-autonomous passive boundary control systems, Evol. Equ. Control Theory, 10 (2021), 385-409.  doi: 10.3934/eect.2020072.

[16]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Birkhäuser, 2012. doi: 10.1007/978-3-0348-0399-1.

[17]

T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan, 5 (1953), 208-234.  doi: 10.2969/jmsj/00520208.

[18]

T. Kato, On linear differential equations in Banach spaces, Comm. Pure Appl. Math., 9 (1956), 479-486.  doi: 10.1002/cpa.3160090319.

[19]

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

[20]

J. Kisyński, Sur les opérateurs de Green des problèmes de Cauchy abstraits, Stud. Math., 23 (1963/64), 285-328.  doi: 10.4064/sm-23-3-285-328.

[21]

M. MiletićD. StürzerA. Arnold and A. Kugi, Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system, IEEE Trans. Autom. Contr., 61 (2016), 2782-2795.  doi: 10.1109/TAC.2015.2499604.

[22]

A. Mironchenko and C. Prieur, Input-to-state stability of infinite-dimensional systems: recent results and open questions, SIAM Rev., 62 (2020), 529-614.  doi: 10.1137/19M1291248.

[23]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Contr., 63 (2018), 1602-1617.  doi: 10.1109/tac.2017.2756341.

[24]

G. Nickel, Evolution semigroups and product formulas for nonautonomous Cauchy problems, Math Nachr., 212 (2000), 101-116.  doi: 10.1002/(SICI)1522-2616(200004)212:1<101::AID-MANA101>3.0.CO;2-3.

[25]

G. Nickel and R. Schnaubelt, An extension of Kato's stability condition for non-autonomous Cauchy problems, Taiwanese J. Math., 2 (1998), 483-496.  doi: 10.11650/twjm/1500407019.

[26]

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

[27]

J. Prüß, On semilinear evolution equations in Banach spaces, J. Reine Angew. Math., 303 (1978), 144-158.  doi: 10.1515/crll.1978.303-304.144.

[28]

J. Prüß, A note on strict solutions to semilinear evolution equations, Math. Z., 171 (1980), 285-288.  doi: 10.1007/BF01214993.

[29]

W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, 1991.

[30]

J. Schmid, Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar, J. Evol. Equ., 16 (2016), 21-50.  doi: 10.1007/s00028-015-0291-5.

[31]

J. Schmid, Adiabatic Theorems for General Linear Operators and Well-Posedness of Linear Evolution Equations, PhD thesis, Universität Stuttgart, 2015. doi: 10.18419/opus-5178.

[32]

J. Schmid, Weak input-to-state stability: Characterizations and counterexamples, Math. Contr. Sign. Syst., 31 (2019), 433-454.  doi: 10.1007/s00498-019-00248-5.

[33]

J. Schmid, Stabilization of port-Hamiltonian systems with discontinuous energy densities, Evol. Equ. Contr. Th., (2022). doi: 10.3934/eect.2021063.

[34]

J. Schmid, S. Dashkovskiy, B. Jacob and H. Laasri, Well-posedness of non-autonomous semilinear systems, Conference Proceedings of the 11th Symposium on Nonlinear Control Systems, IFAC-PapersOnLine, 52 (2019), 216–220. doi: 10.1016/j.ifacol.2019.11.781.

[35]

J. Schmid and M. Griesemer, Kato's theorem on the integration of non-autonomous evolution equations, Math. Phys. Anal. Geom., 17 (2014), 265-271.  doi: 10.1007/s11040-014-9154-5.

[36]

J. Schmid, O. Kapustyan and S. Dashkovskiy, Asymptotic gain results for attractors of semilinear systems, Math. Contr. Rel. Fields, (2021). doi: 10.3934/mcrf.2021044.

[37]

J. Schmid and M. Griesemer, Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces, Math. Nachr., 290 (2017), 435-441.  doi: 10.1002/mana.201500052.

[38]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances., ESAIM Contr. Optim. Calc. Var., 27 (2021), Paper No. 53, 37 pp. doi: 10.1051/cocv/2021051.

[39]

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

[40]

R. Schnaubelt and G. Weiss, Two classes of passive time-varying well-posed linear systems, Math. Contr. Sign. Syst., 21 (2010), 265-301.  doi: 10.1007/s00498-010-0049-0.

[41]

F. L. Schwenninger, Input-to-state stability for parabolic boundary control: Linear and semi-linear systems, Control Theory of Infinite-Dimensional Systems, Operator Theory: Advances and Applications, 277 (2020), 83-116.  doi: 10.1007/978-3-030-35898-3_4.

[42]

N. Skrepek, Well-posedness of linear first order port-Hamiltonian systems on multidimensional spatial domains, Evol. Equ. Control Theory, 10 (2021), 965-1006.  doi: 10.3934/eect.2020098.

[43]

M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Contr. Sign. Syst., 2 (1989), 265-285.  doi: 10.1007/BF02551387.

[44]

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

[45]

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.

[46]

A. van der Schaft, $L^2$-Gain and Passivity Techniques in Nonlinear Control, 3rd edition, Springer, 2017. doi: 10.1007/978-3-319-49992-5.

[47]

J. C. Willems, Dissipative dynamical systems. Part I: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351.  doi: 10.1007/BF00276493.

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