Input-output stability degrees for undamped constant coefficients linear partial differential equations

It is an established fact that systems which have transfer matrices with po les converging to the imaginary axis cannot have exponentially stable time responses. Recently it was proved that for a class of partial differential equations with such a structure of poles it is possible to have a fast deca y in time, uniformly in space, as arbitrary polynomials if the initial cond itions are smooth enough and with an appropriate decay at infinity. The non -homogeneous version of this result which we present here can be summarized as follows: 'arbitrary regularity in space of the input function leads to arbitrary polynomial convergence in time towards the steady state'. The rel ation between the properties of the input function and the rate of converge nce of the poles to the imaginary axis is quantitative and we indicate meth ods for computing this rate. We also provide conditions for exponential sta bility in this context. Due to some limitations in exponential stabilization by feedback a natural alternative to stability enhancement is this polynomial one. Therefore it i s useful to investigate when it can be recovered in more practical situatio ns (bounded space, boundary control, etc). Possible applications include th e control of distributed oscillatory phenomena (e. g. in large flexible str uctures, plates), and more recently the control of some advanced materials.