Nehari problems and equalizing vectors for infinite-dimensional systems

For a class of infinite-dimensional systems we obtain a simple frequency domain solution for the suboptimal Nehari extension problem. The approach is via $J$-spectral factorization, and it uses the concept of equalizing vectors. Moreover, the connection between the equalizing vectors and the Nehari extension problem is given. \u

J-spectral factorization and equalizing vectors

For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a $J$-spectral factorization. One of these conditions is in terms of equalizing vectors. A second one states that the existence of a $J$-spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory. Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard-Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given. For Riesz-spectral systems another necessary and sufficient conditions for the existence of a $J$-spectral factorization in terms of the Hamiltonian is added

Riesz spectral systems

In this paper we study systems in which the system operator, $A$, has a Riesz basis of (generalized) eigenvectors. We show that this class is subset of the class of spectral operators as studied by Dunford and Schwartz. For these systems we investigate several system theoretic properties, like stability and controllability. We apply our theory to Euler-Bernoulli beam with structural damping

Sufficient Conditions for Admissibility

Sufficient conditions for the finite and infinite-time admissibility of an observation operator are given. It is shown that the estimates of Weiss are close to being sufficient. If the semigroup is surjective, then the estimate is sufficient

Climate control of a bulk storage room for foodstuffs

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

Dirac structures and boundary control systems associated with skew-symmetric differential operators

Associated with a skew-symmetric linear operator on the spatial domain $[a,b]$ we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the $C_{0}$-semigroup associated to this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko Beam. \u