Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body

• Published in: Mathematics of control, signals, and systems Vol. 33; no. 4; pp. 637 - 667
• Main Authors: Mitra, Debanjana, Roy, Arnab, Takahashi, Takéo
• Format: Journal Article
• Language: English
• Published: London Springer London 01.12.2021; Springer Nature B.V; Springer Verlag
• Subjects: Communications Engineering; Control; Controllability; Feedback control; Fluid-structure interaction; Linearization; Mathematics; Mathematics and Statistics; Mechatronics; Networks; Original Article; Rigid structures; Robotics; Stability; Systems Theory; Viscoelastic fluids; Viscoelasticity; Stabilizability; 76A10; 93D15; 35Q35; 74F10; Controllability; Fluid–structure interaction systems; Viscoelastic fluids; Finite-dimensional controls; 93B52; controllability; Fluid-structure interaction systems; stabilizability; viscoelastic fluids; finite dimensional controls 2010 Mathematics Subject Classification 76A10
• ISSN: 0932-4194; 1435-568X
• DOI: 10.1007/s00498-021-00295-x

We study control properties of a linearized fluid–structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that as in the case without structure, the system is not exactly null-controllable in finite time.