A dynamic system with structural damping described by partial differential equations is investigated. The system is first converted to an abstract evolution equation in an appropriate Hilbert space, and the spectral and semigroup properties of the system operator are discussed. Finally, the well-posedness and the asymptotical stability of the system are obtained by means of a semigroup of linear operators.
The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix A^{T}PA-P is negative definite. The theoretical deliberations will be illustrated by numerical examples.
A trajectory tracking problem for the three-dimensional kinematic model of a unicycle-type mobile robot is considered. It is assumed that only two of the tracking error coordinates are measurable. By means of cascaded systems theory we develop observers for each of the error coordinates and show the K-exponential convergence of the tracking error in combined closed-loop observer-controller systems. The results are illustrated with computer simulations.