Roberto Triggiani

On the stabilizability problem in Banach space

Abstract The model is a linear system defined on Banach (state and control) spaces, with the operator acting on the state only the infinitesimal generator of a strongly continuous semigroup. The stabilizability problem of expressing the control through a bounded operator acting on the state as to make the resulting feedback system globally asymptotically stable is considered. On the negative side, and in contrast with the finite dimensional theory, a few counter examples are given of systems which are densely controllable in the space and yet are not stabilizable, even if some further “nice properties” hold. Use is made of the notion of essential spectrum and its stability under relatively compact perturbations. On the positive side, it is shown, however, that for large classes of systems of physical interest (classical selfadjoint boundary value problems, delay equations, etc.) controllability on a suitable finite dimensional subspace still yields stabilizability on the whole space.

A Note on the Lack of Exact Controllability for Mild Solutions in Banach Spaces

It is shown that exact controllability in finite time for linear control systems given on an infinite dimensional Banach space in integral form (mild solution) can never arise using locally

Function Space Controllability of Linear Retarded Systems: A Derivation from Abstract Operator Conditions

L_1

Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems

-controls, if the associated

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

C_0

Controllability and Observability in Banach Space with Bounded Operators

semigroup is compact for all

Boundary feedback stabilizability of parabolic equations

t > 0

A cosine operator approach to modelingL2(0,T; L2 (Γ))--Boundary input hyperbolic equations

. This includes, in particular, the class of parabolic partial differential equations defined on bounded spatial domains.

Extensions of Rank Conditions for Controllability and Observability to Banach Spaces and Unbounded Operators

Controllability of linear retarded systems is investigated by using the abstract representation of such systems given by

Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications

\dot x = \tilde Ax + \tilde Bu