Controllability of Linear and Nonlinear Control Systems related through Simulation Relations

For nonlinear input-disturbance systems that are connected by a simulation relation, we examine to what extent they share certain controllability properties. Specifically our main objective is to determine the conditions under which the following holds: given two control systems $F$ and $\tilde{F}$ where $F$ is simulated by $\tilde{F}$ and $F$ is completely controllable, we have that $\tilde{F}$ is also completely controllable. To this end, we show that under some additional conditions the property of complete controllability is preserved for pointwise graph simulation relations and compact graph simulation relations. Next in an attempt to prove a similar result between a nonlinear system and an almost linear system, but with the simulation relation submanifold being a regular level set of a particular map instead of a graph, we achieve the result of the simulating system $\tilde{F}$ being at most completely controllable modulo the kernel of a linear map. We show through an example that $\tilde{F}$ may fail to be completely controllable if it does not fulfill a certain compactness condition. By imposing this compactness condition along with other somewhat restrictive assumptions, we are able to prove a similar result for nonlinear control systems connected through a simulation relation submanifold in the form of a regular level set of a smooth mapping. We then illustrate the features of our final main result with an example

Geometric control of particle manipulation in a two-dimensional fluid

Manipulation of particles suspended in fluids is crucial for many applications, such as precision machining, chemical processes, bio-engineering, and self-feeding of microorganisms. In this paper, we study the problem of particle manipulation by cyclic fluid boundary excitations from a geometric-control viewpoint. We focus on the simplified problem of manipulating a single particle by generating controlled cyclic motion of a circular rigid body in a two-dimensional perfect fluid. We show that the drift in the particle location after one cyclic motion of the body can be interpreted as the geometric phase of a connection induced by the system's hydrodynamics. We then formulate the problem as a control system, and derive a geometric criterion for its nonlinear controllability. Moreover, by exploiting the geometric structure of the system, we explicitly construct a feedback-based gait that results in attraction of the particle towards the rigid body. We argue that our gait is robust and model-independent, and demonstrate it in both perfect fluid and Stokes fluid

Generation of two-dimensional water waves by moving bottom disturbances

We investigate the potential and limitations of the wave generation by disturbances moving at the bottom. More precisely, we assume that the wavemaker is composed of an underwater object of a given shape which can be displaced according to a prescribed trajectory. We address the practical question of computing the wavemaker shape and trajectory generating a wave with prescribed characteristics. For the sake of simplicity we model the hydrodynamics by a generalized forced Benjamin-Bona-Mahony (BBM) equation. This practical problem is reformulated as a constrained nonlinear optimization problem. Additional constraints are imposed in order to fulfill various practical design requirements. Finally, we present some numerical results in order to demonstrate the feasibility and performance of the proposed methodology.Comment: 21 pages, 7 figures, 1 table, 69 references. Other author's papers can be downloaded at http://www.denys-dutykh.com