A note on polynomially growing C-0-semigroups

We characterize polynomial growth of a $C_0$-semigroup in terms of the first power of the resolvent of its generator. We do this for a class of semigroups which includes $C_0$-semigroups on Hilbert spaces and analytic semigroups on Banach spaces. Furthermore, we characterize polynomial growth for discrete semigroups

The duality between the gradient and divergence operators on bounded Lipschitz domains

This report gives an exact result on the duality of the divergence and gradient operators, when these are considered as operators between $L^2$-spaces on a bounded $n$-dimensional Lipschitz domain. The necessary background is described in detail, with a self-contained exposition

The growth of a semigroup and its Cayley transform

Let $A$ be the infinitesimal generator of an exponentially stable, strongly continuous semigroup on a Hilbert space. We show that the powers of the Cayley transform of $A$ are bounded by a constant times $\log (n+1)$. The proof is based on Lyapunov equations

Boundary control for a class of dissipative differential operators including diffusion systems

In this paper we study a class of partial differential equations (PDE's), which includes Sturm-Liouville systems and diffusion equations. From this class of PDE's we define systems with control and observation through the boundary of the spatial domain. That is, we describe how to select boundary conditions, such that the resulting system has inputs and outputs acting through the boundary. Furthermore, these boundary conditions are chosen in a way that the resulting system has a nonincreasing energy.\u

Controllability and stability of 3D heat conduction equation in a submicroscale thin film

We obtain a closed form analytic solution for the Dual Phase Lagging equation. This equation is a linear, time-independent partial differential equation modeling the heat distribution in a thin film. The spatial domain is of micrometer and nanometer geometries. We show that the solution is described by a semigroup, and obtain a basis of eigenfunctions. The closure of the set of eigenvalues contains an interval, and so the theory on Riesz spectral operator of Curtain and Zwart cannot be applied directly. The exponential stability and the approximate controllability is shown

Analytic control law for a food storage room

A storage room contains a bulk of potatoes that produce heat due to respiration. A ventilator blows cooled air around to keep the potatoes cool and prevent spoilage. The aim is to design a control law such that the product temperature is kept at a constant, desired level. This physical system is modelled by a set of nonlinear coupled partial differential equations (pde's) with nonlinear input. Due to their complex form, standard control design will not be adequate. A novel modelling procedure is proposed. The input is considered to attain only discrete values. Analysis of the transfer functions of the system in the frequency domain leads to a simplification of the model into a set of static ordinary differential equations ode's). The desired control law is now the optimal time to switch between the discrete input values on an intermediate time interval. The switching time can be written as a symbolic expression of all physical parameters of the system. Finally, a dynamic controller can be designed that regulates the air temperature on a large time interval, by means of adjustment of the switching time