In recent years, the stability and lifetime of nanobubbles have raised new scientific ques- tions as experimental techniques for nanobubble generation and measurements have ad- vanced. In this contribution, we present a physical model for bulk nanobubbles and sim- ulate their dissolution numerically using a rigorous treatment of the underlying governing equations. Additionally, we simulate surface nanobubbles with small contact angles replen- ished through hydrogen electrolysis at a platinum electrode, and discuss the relation be- tween the mass transfer coefficient at the bubble surface and the supplied hydrogen flux at steady-state.

We consider an elastic structure that is subject to moving loads representing e.g. heavy trucks on a bridge or a trolley on a crane beam. A model for the quasi-static mechanical behaviour of the structure is derived, yielding a coupled problem involving partial differential equations (PDE) and ordinary differential equations (ODE). The problem is simulated numerically and validated by comparison with a standard formula used in engineering. We derive an optimal policy for passing over potentially fragile bridges. In general, our problem class leads to optimal control problems subject to coupled ODE and PDE.

In recent years, the stability and lifetime of nanobubbles have raised new scientific ques- tions as experimental techniques for nanobubble generation and measurements have ad- vanced. In this contribution, we present a physical model for bulk nanobubbles and sim- ulate their dissolution numerically using a rigorous treatment of the underlying governing equations. Additionally, we simulate surface nanobubbles with small contact angles replen- ished through hydrogen electrolysis at a platinum electrode, and discuss the relation be- tween the mass transfer coefficient at the bubble surface and the supplied hydrogen flux at steady-state.

In this article necessary optimality conditions for Saint-Venant equations coupled to ordinary differential equations (ODE) are derived rigorously. The Saint-Venant equations are first-order hyperbolic partial differential equations (PDE) and model here the fluid in a container that is moved by a truck that is subject to Newton's law of motion. The acceleration of the truck may be controlled. We describe the mathematical model and the corresponding tracking-type optimal control problem. First we prove existence and uniqueness for the coupled ODE-PDE problem locally in time. For sufficiently small times, we derive the first-order necessary optimality conditions in the corresponding function spaces. Furthermore we prove existence of optimal controls. The optimality system is formulated in the setting of a semi-smooth operator equation in Hilbert spaces which we solve numerically by a semi-smooth Newton method. We close with a numerical example for a typical driving maneuver.

We present a mathematical model of a crane-trolley-load model, where the crane beam is subject to the partial differential equation (PDE) of static linear elasticity and the motion of the load is described by the dynamics of a pendulum that is fixed to a trolley moving along the crane beam. The resulting problem serves as a case study for optimal control of fully coupled partial and ordinary differential equations (ODEs). This particular type of coupled systems arises from many applications involving mechanical multi-body systems. We motivate the coupled ODE-PDE model, show its analytical well-posedness locally in time and examine the corresponding optimal control problem numerically by means of a projected gradient method with Broyden-Fletcher-Goldfarb-Shanno (BFGS) update.

In this study we present an extension of a model of an elastic crane transporting a load by means of controlling the crane trolley motion and the crane rotation. In addition to the model considered in Kimmerle et al. (2017), we allow for rotations of the crane and include damping of the trolley and moments of inertia as well. We derive a fully coupled system of ordinary differential equations (ODE), representing the trolley and load (modelled as a pendulum), and partial differential equations (PDE), i.e. the linear elasticity equations for the deformed crane beam. The objective to be minimized is a linear combination of the terminal time, the control effort, the kinetic energy of the load, and penalty terms for the terminal conditions. We show the Fréchet-differentiability of the mechanical displacement field with respect to the location of the boundary condition that is moving. This is a crucial point for a further mathematical analysis on the existence of optimal controls and the derivation of necessary optimality conditions. Finally we present first results for the full time-optimal control of the extended model.

We consider a mathematical model for surface nanobubbles arising from hydrogen electrolysis in polymer electrolyte mem- brane (PEM) electrolyzers. Experimental advances in recent years indicated longer lifetimes of surface nanobubbles than may be explained by classical theories. Contrary to [5], we state a full model of an evolving single surface nanobubble yielding a coupled system consisting of a partial differential equation (PDE) for the hydrogen concentration in water and an ordinary differential equation (ODE) for the radius evolution. In the special case of dynamic equilibrium conditions, we prove the well-posedness of this steady state problem by a fixed-point strategy, assuming an acute-angled contact angle, and that the corresponding algorithm allows for its numerical simulation.