Analysis of stabilization operators in a Galerkin least-squares finite element discretization of the incompressible Navier-Stokes equations

In this paper the design and analysis of a dimensionally consistent stabilization operator for a time-discontinuous Galerkin least-squares finite element method for unsteady viscous flow problems governed by the incompressible Navier-Stokes equations, is discussed. The analysis results in a class of stabilization operators which satisfy essential conditions for the stability of the numerical discretization

Photothermal effects in ultra-precisely stabilized tunable microcavities

We study the mechanical stability of a tunable high-finesse microcavity under ambient conditions and investigate light-induced effects that can both suppress and excite mechanical fluctuations. As an enabling step, we demonstrate the ultra-precise electronic stabilization of a microcavity. We then show that photothermal mirror expansion can provide high-bandwidth feedback and improve cavity stability by almost two orders of magnitude. At high intracavity power, we observe self-oscillations of mechanical resonances of the cavity. We explain the observations by a dynamic photothermal instability, leading to parametric driving of mechanical motion. For an optimized combination of electronic and photothermal stabilization, we achieve a feedback bandwidth of $500\,$kHz and a noise level of $1.1 \times 10^{-13}\,$m rms

Asymptotic Stability, Instability and Stabilization of Relative Equilibria

In this paper we analyze asymptotic stability, instability and stabilization for the relative equilibria, i.e. equilibria modulo a group action, of natural mechanical systems. The practical applications of these results are to rotating mechanical systems where the group is the rotation group. We use a modification of the Energy-Casimir and Energy-Momentum methods for Hamiltonian systems to analyze systems with dissipation. Our work couples the modern theory of block diagonalization to the classical work of Chetaev