Stability and control of systems with propagation phenomena

There are considered controlled systems with distributed parameters in one dimension. Their mathematical model is given by mixed initial boundary value problems for hyperbolic partial differential equations in two dimensions. The control is performed at the boundaries, being described by ordinary differential equations accounting for the dynamics of the controllers. Two are the standard problems for such systems, especially in the nonlinear case: well posedness in the sense of J. Hadamard (existence,uniqueness and smooth dependence on data) and stability of the steady state solutions - a control oriented property. While feedback control synthesis may be performed at the formal level, validation of the resulting closed loop model as well as stability have to be analyzed with the mathematical rigor required by the theory of partial differential equations. The present paper is focused on these topics starting from the basic model of the flexible beam, accounting for the models of the flexible manipulator arm, of the overhead crane and of the oilwell drillstring. It is shown how various approaches that are known from the fundamentals of partial differential equations can be used to obtain the necessary properties.