Optimal input-output stabilization of infinite-dimensional discrete time-invariant linear systems by output injection

We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by output injection. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a left factorization over H-infinity. Another equivalent condition is that the filter Riccati equation (of an arbitrary realization) has a solution (in general, unbounded and even nondensely defined). We further show that after renorming the state space in terms of the inverse of the smallest solution of the filter Riccati equation, the closed-loop system is not only input-output stable but also strongly internally *-stable

Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator

We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state. We give necessary and sufficient condition to have an exponential rate of convergence. Numerical simulations are presented to illustratethe theoretical results

Volterra integral and functional equations

This book looks at the theories of Volterra integral and functional equations

Absolute-stability results in infinite dimensions

We derive absolute-stability results of Popov and circle-criterion type for infinite-dimensional systems in an input-output setting. Our results apply to feedback systems in which the linear part is the series interconnection of an input-output stable linear system and an integrator, and the nonlinearity satisfies a sector condition which, in particular, allows for saturation and deadzone effects. We use the input-output theory developed to derive state-space results on absolute stability applying to feedback systems in which the linear part is the series interconnection of an exponentially stable, well-posed infinite-dimensional system and an integrator

Stability results of Popov-type for infinite-dimensional systems with applications to integral control

We derive absolute stability results of Popov-type for infinite-dimensional systems in an input-output setting. Our results apply to feedback systems where the linear part is the series interconnection of an L2-stable linear system and an integrator, and the non-linearity satisfies a sector condition which allows for non-linearities with lower gain equal to zero (such as saturation, or more generally, bounded non-linearities). These results are used to prove convergence and stability properties of low-gain integral feedback control applied to L2-stable linear systems subject to actuator and sensor non-linearities. The class of actuator/sensor non-linearities under consideration contains standard non-linearities which are important in control engineering such as saturation and deadzone. Moreover, we use the input-output theory developed to derive state-space results on absolute stability and low-gain integral control for strongly stable well-posed infinite-dimensional linear systems. 2000 Mathematics Subject Classification 45M05, 45M10, 93B52, 93C10, 93C20, 93C25, 93D05, 93D09, 93D10, 93D25