The fluid-structure interaction model with both control and disturbance at the interface: a game theory problem via an abstract approach

Abstract

We consider an established hyperbolic–parabolic fluid-structure interaction model with control and disturbance acting at the interface between the two media. Here, the structure (modelled by the system of dynamic elasticity) is immersed in a fluid (modelled by the linearized Navier–Stokes equations). A game theory problem between control and disturbance is studied, when both act at the interface. To this end, the main mathematical difficulty that one encounters is the fact that such model fails to satisfy the ‘singular estimate’ from, say, control to state. This is a critical obstacle, as this is precisely the foundational property used in the development by Triggiani and Zhang [Min–max game theory and non-standard differential Riccati equations under singular estimates for e ^At B and e ^At G in the absence of analyticity, Set-Valued Var. Anal., 17 (2009), pp. 245–283.] for a full theory to include the solvability of the Differential Riccati equation in the study of the associated min–max game theory problem. Failure of the ‘singular estimate’ property is due to a mismatch between the parabolic and hyperbolic component of the overall coupled dynamics. Next, by introducing suitable observation or output operators, with incremental smoothness on the trajectory, it is shown that the resulting system satisfies a modified singular estimate, this time from the control to the observation space. This then allows one to adapt (and generalize) the complete min–max theory by Triggiani and Zhang to the present fluid-structure interaction model. More precisely, the approach followed is based on an abstract setting, of which the fluid-structure interaction model is a canonical illustration, which enjoys a desirable property not assumed for the abstract model.

Keywords:

• fluid structure
• interface
• game theory

AMS Subject Classifications:

• 35K20
• 35L20
• 25Q30
• 49J20
• 76B03
• 93B36
• 93C20

Acknowledgements

Research by I. Lasiecka and R. Triggiani was partially supported by the National Science Foundation under grant DMS-0104305 and the Air Force Office of Scientific Research under grant FA 9550-09-1-0459.