Abstract. This paper is concerned with the stability analysis of a lossless Euler-Bernoulli beam that carries a tip payload which is coupled to a nonlinear dynamic feedback system. This setup comprises nonlinear dynamic boundary controllers satisfying the nonlinear KYP lemma as well as the interaction with a nonlinear passive environment. Global-in-time wellposedness and asymptotic stability is rigorously proven for the resulting closed-loop PDE-ODE system. The analysis is based on semigroup theory for the corresponding first order evolution problem. For the large-time analysis, precompactness of the trajectories is shown by deriving uniform-in-time bounds on the solution and its time derivatives.

In this paper, we analyze a systematically designed and easily tunable backstepping-based boundary control concept developed by Thull, Wild, and Kugi (2006) for a gantry crane with heavy chain and payload. The corresponding closedloop system is formulated as an abstract evolution equation in an appropriate Hilbert space. Non-restrictive conditions for the controller coefficients are derived, under which the solutions are described by a C0-semigroup of contractions, and are asymptotically stable. Moreover, by applying Huang's theorem we can finally even show that under these conditions the controller renders the closed-loop system exponentially stable.Date: October 1, 2018.

Abstract. In this article we consider a Fokker-Planck equation on R d with a non-local, mass preserving perturbation. We first give a spectral analysis of the unperturbed Fokker-Planck operator in an exponentially weighted L 2 -space. In this space the perturbed Fokker-Planck operator is an isospectral deformation of the Fokker-Planck operator, i.e. the spectrum of the Fokker-Planck operator is not changed by the perturbation. In particular, there still exists a unique (normalized) stationary solution of the perturbed evolution equation. Moreover, the perturbed Fokker-Planck operator generates a strongly continuous semigroup of bounded operators. Any solution of the perturbed equation converges towards the stationary state with exponential rate −1, the same rate as for the unperturbed Fokker-Planck equation. Moreover, for any k ∈ N there exists an invariant subspace with codimension k (if d = 1) in which the exponential decay rate of the semigroup equals −k.

We study the asymptotic behavior for a system consisting of a clamped flexible beam that carries a tip payload, which is attached to a nonlinear damper and a nonlinear spring at its end. Characterizing the ω-limit sets of the trajectories, we give a sufficient condition under which the system is asymptotically stable. In the case when this condition is not satisfied, we show that the beam deflection approaches a non-decaying time-periodic solution.2010 Mathematics Subject Classification. 35B40, 70K20, 74K10, 47H20, 35Q70.

We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for nonsymmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with an uniform rate.This research was partially supported by the FWF-doctoral school "Dissipation and dispersion in nonlinear partial differential equations" and INDAM -GNFM from Italy. One author (AA) is grateful to J. Schöberl for very helpful discussions.