Well-posedness and exact controllability of fourth-order Schrödinger equation with hinged boundary control and collocated observation

In this paper, we consider the well-posedness and exact controllability of a fourth-order multi-dimensional Schrödinger equation with hinged boundary by either moment or Dirichlet boundary control and collocated observation, respectively. It is shown that in both cases, the systems are well posed in the sense of D. Salamon, which implies that the systems are exactly controllable in some finite time interval if and only if its corresponding closed loop systems under the direct output proportional feedback are exponentially stable. This leads us to discuss further the exact controllability of the systems. In addition, the systems are consequently shown to be regular in the sense of G. Weiss as well, and the feedthrough operators are zero.

[1]  Benoit Pausader,et al.  Global Well-Posedness for Energy Critical Fourth-Order Schrodinger Equations in the Radial Case , 2007 .

[2]  Ruth F. Curtain,et al.  The Salamon—Weiss class of well-posed infinite-dimensional linear systems: a survey , 1997 .

[3]  Kaïs Ammari,et al.  Stabilization of second order evolution equations by a class of unbounded feedbacks , 2001 .

[4]  L. Hsiao,et al.  Wellposedness for the fourth order nonlinear Schrödinger equations , 2006 .

[5]  Bao-Zhu Guo,et al.  Regularity of an Euler-Bernoulli Equation with Neumann Control and Collocated Observation , 2006 .

[6]  R. Triggiani,et al.  The operator $B^*L$ for the wave equation with Dirichlet control , 2004 .

[7]  Thorsten Gerber,et al.  Semigroups Of Linear Operators And Applications To Partial Differential Equations , 2016 .

[8]  P. Grisvard,et al.  Caractérisation de quelques espaces d'interpolation , 1967 .

[9]  George Weiss,et al.  Admissibility of unbounded control operators , 1989 .

[10]  V. Komornik.  Exact Controllability and Stabilization: The Multiplier Method , 1995 .

[11]  Dietmar A. Salamon,et al.  Realization theory in Hilbert space , 1988, Mathematical systems theory.

[12]  Shugen Chai,et al.  Well-posedness and exact controllability of fourth-order Schrödinger equation with hinged boundary control and collocated observation , 2014, Mathematics of Control, Signals, and Systems.

[13]  J. Lions.  Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[14]  Hans Zwart,et al.  Transfer functions for infinite-dimensional systems , 2004, Syst. Control. Lett..

[15]  Bao-Zhu Guo,et al.  Feedthrough Operator for Linear Elasticity System with Boundary Control and Observation , 2010, SIAM J. Control. Optim..

[16]  Kaïs Ammari,et al.  Dirichlet boundary stabilization of the wave equation , 2002 .

[17]  Ruth F. Curtain,et al.  Dynamic stabilization of regular linear systems , 1997, IEEE Trans. Autom. Control..

[18]  Mark R. Opmeer,et al.  Infinite‐Dimensional Linear Systems: a Distributional Approach , 2005 .

[19]  Benoit Pausader,et al.  The cubic fourth-order Schrödinger equation , 2008, 0807.4916.

[20]  Bao-Zhu Guo,et al.  Regularity of a Schrödinger equation with Dirichlet control and colocated observation , 2005, Syst. Control. Lett..

[21]  Ruth F. Curtain,et al.  Well posedness of triples of operators (in the sense of linear systems theory) , 1989 .

[22]  Olof J. Staffans,et al.  A Physically Motivated Class of Scattering Passive Linear Systems , 2012, SIAM J. Control. Optim..

[23]  Irena Lasiecka,et al.  Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only , 1992 .

[24]  Ruth F. Curtain,et al.  Exponential stabilization of well-posed systems by colocated feedback , 2006, SIAM J. Control. Optim..

[25]  Bao-Zhu Guo,et al.  Well-posedness and regularity of weakly coupled wave-plate equation with boundary control and observation , 2009 .

[26]  Marius Tucsnak,et al.  Well-posed linear systems a survey with emphasis on conservative systems , 2001 .

[27]  Bao-Zhu Guo,et al.  Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator , 2002, Syst. Control. Lett..

[28]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[29]  Ruth F. Curtain,et al.  Linear Operator Inequalities for Strongly Stable Weakly Regular Linear Systems , 2001, Math. Control. Signals Syst..

[30]  L. Hörmander.  Linear Partial Differential Operators , 1963 .

[31]  V. Karpman,et al.  Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion , 2000 .

[32]  R. Triggiani,et al.  L2(Σ)-regularity of the boundary to boundary operator B∗L for hyperbolic and Petrowski PDEs , 2003 .

[33]  Christopher I. Byrnes,et al.  Regular Linear Systems Governed by a Boundary Controlled Heat Equation , 2002 .

[34]  George Weiss,et al.  Admissible observation operators for linear semigroups , 1989 .

[35]  L. Hsiao,et al.  Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces , 2007 .

[36]  O. Staffans.  Well-Posed Linear Systems , 2005 .

[37]  Xu Zhang,et al.  The Regularity of the Wave Equation with Partial Dirichlet Control and Colocated Observation , 2005, SIAM J. Control. Optim..

[38]  Karpman,et al.  Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[39]  Bao-Zhu Guo,et al.  ON WELL-POSEDNESS, REGULARITY AND EXACT CONTROLLABILITY FOR PROBLEMS OF TRANSMISSION OF PLATE EQUATION WITH VARIABLE COEFFICIENTS , 2007 .

[40]  D. Salamon.  Infinite Dimensional Linear Systems with Unbounded Control and Observation: A Functional Analytic Approach. , 1987 .

[41]  G. Weiss,et al.  Observation and Control for Operator Semigroups , 2009 .

[42]  George Weiss,et al.  Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity , 1994 .

[43]  Chuang Zheng,et al.  Exact controllability for the fourth order Schrödinger equation , 2012 .