- Fokker-Planck equations with nonlocal perturbations: Equations of this type arise in the kinetic formulation of quantum mechanical systems (e.g., the Wigner-Fokker-Planck equation). The Fokker-Planck operator is linear, and the perturbation is the convolution with a massless distribution, satisfying weak regularity assumptions. I did a complete spectral analysis of the perturbed operator in weighted Lebesgue spaces, and gave a detailed analysis of the generated semigroup of operators. The main result is that there exists a unique normalized steady state, and every other solution converges with an explicitly known exponential rate towards this steady state.
- Nonlinear
control of an Euler-Bernoulli beam:
The systems considered here describe nonlinear feedback control
in
order to stabilize vibrations of eleastic beams. Applications
comprise
robotic and hydraulic arms, antennae, and high-rise buildings.
Asymtotic stability of these systems has been shown by my
coauthors and
me by using tools of nonlinear functional analysis, confirming
the
functionality of the controller.

Fig.: An elastic beam with a damper and a spring (from [4]).

- D. Stürzer, A. Arnold, A. Kugi. Stability of a Closed-Loop Control System – Applied to a Gantry Crane with Heavy Chains. Proceedings of the Junior Scientist Conference, (2010).
- D. Stürzer, A. Arnold. Spectral analysis and long-time behaviour of a Fokker-Planck equation with a nonlocal perturbation. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25, 1 (2014), 53–89.
- F. Achleitner, A. Arnold, D. Stürzer. Large-time behavior in nonsymmetric Fokker-Planck equations. Rivista di Matematica della Università di Parma 6, 1 (2015).
- M. Miletic, D. Stürzer, A. Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete and Continuous Dynamical System - B 20, 9 (2015).
- M. Miletic, D. Stürzer, A. Arnold, A. Kugi. Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system. IEEE Transactions on Automatic Control 61, 10 (2016).
- D. Stürzer, A. Arnold, and A. Kugi. Closed-loop stability analysis of a gantry crane with heavy chain. To appear in International Journal of Control (2017).