Steady-state gains and sample-hold discretisations of infinite-dimensional linear systems

At the basis of our considerations are the algebra of shift-invariant bounded linear input-output operators on L²(Ropf₊) and the subalgebra consisting of all convolution operators with measure kernels together with their discrete-time counterparts. We introduce the concepts of asymptotic steady-state gain, L²-steady-state gain and l²-steady-state gain. We give some natural conditions under which the existence of these steady-state gains is guaranteed. Under a mild assumption on the continuous-time system it is shown that the existence of the (continuous-time) L²-steady-state gain implies the existence of the l²-steady-state gain of the sample-hold discretisation and that these two gains coincide. The sample and hold operations considered are ideal sampling, generalised sampling and zero-order hold. Applications to sampled-data control are briefly discussed