Feedback control of charged ideal fluids

Abstract

The theory of controlled mechanical systems of Bloch et al (2001 IEEE Trans. Control Syst. Technol. 45 2253–70; Bloch et al 2001 Int. J. Robust Nonlinear Control 11 191–214; Bloch et al 1992 Automatica 28 745–56; Bloch et al 1997 Current and Future Directions in Applied Mathematics) and Krishnaprasad P S (1985 Nonlinear Anal. Theor. Methods Appl. 9 1011–35) is extended to the case of ideal incompressible fluids consisting of charged particles in the presence of an external magnetic field. The resulting control is of feedback type and depends on the Eulerian state of the controlled system. Moreover, the control is set up so that the corresponding closed loop equations are Lie–Poisson. This implies that the energy–momentum method of Arnold V (1966 Ann. Inst. Fourier 16 319–61) and Holm et al (1985 Phys. Rep. 123 1–116) can be used to find a stabilizing control. As an example the case of planar parallel shear flow with an inflection point is treated. A state dependent feedback control is constructed which stabilizes the system for an arbitrarily long channel.