Well-Posedness and Controllability of a Class of Coupled Linear Systems

We consider coupled systems consisting of an infinite-dimensional part and a finite-dimensional part connected in feedback, such as, for example, the well-known SCOLE system (a beam with a rigid body attached at one end). The external world interacts with the coupled system via the finite-dimensional part, which receives the external input and sends out the output. The infinite-dimensional part is assumed to be such that it becomes well-posed and strictly proper when connected in cascade with an integrator. Under several assumptions, we derive well-posedness and exact controllability results for such coupled systems. The first main result concerns the case when the input signal of the finite-dimensional part is the difference between the external input and the feedback signal. The second main result allows a more general structure for the finite-dimensional part. We also prove a result for the approximate controllability of coupled systems.

[1]  I. Lasiecka.  Mathematical control theory of couple PDEs , 2002 .

[2]  B. Guo,et al.  Boundary Controllability and Observability of a One-Dimensional Nonuniform SCOLE System , 2005 .

[3]  Olof J. Staffans.  Passive and Conservative Continuous-Time Impedance and Scattering Systems. Part I: Well-Posed Systems , 2002, Math. Control. Signals Syst..

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

[5]  B. Rao.  Exact boundary controllability of a hybrid system of elasticity by the HUM method , 1997 .

[6]  George Weiss,et al.  Regular linear systems with feedback , 1994, Math. Control. Signals Syst..

[7]  Yuri Latushkin,et al.  Regularization and frequency-domain stability of well-posed systems , 2005, Math. Control. Signals Syst..

[8]  Walter Littman,et al.  Exact boundary controllability of a hybrid system of elasticity , 1988 .

[9]  D. L. Russell.  Review: J.-L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués , 1990 .

[10]  J. Malinen,et al.  Impedance Passive and Conservative Boundary Control Systems , 2007 .

[11]  C. SIAMJ..  SIMULTANEOUS EXACT CONTROLLABILITY AND SOME APPLICATIONS , 2000 .

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

[13]  Olof J. Staffans,et al.  Transfer Functions of Regular Linear Systems Part III: Inversions and Duality , 2004 .

[14]  J. A. Villegas,et al.  A Port-Hamiltonian Approach to Distributed Parameter Systems , 2007 .

[15]  Bao-Zhu Guo,et al.  On the Boundary Control of a Hybrid System with Variable Coefficients , 2002 .

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

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

[18]  George Weiss,et al.  Strong Stabilization of a Non-uniform SCOLE Model , 2008 .

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

[20]  Ramkrishna Pasumarthy.  On analysis and control of interconnected finite- and infinite-dimensional physical systems , 2006 .

[21]  Olof J. Staffans,et al.  Transfer functions of regular linear systems. Part II: The system operator and the lax-phillips semigroup , 2002 .

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

[23]  K. G. Budden.  Wave Propagation , 1969, Nature.

[24]  L. Markus,et al.  Stabilization of a hybrid system of elasticity by feedback boundary damping , 1988 .

[25]  Enrique Zuazua,et al.  Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures (Mathématiques et Applications) , 2005 .