This paper studies the stabilization for a kind of linear and impulse control systems in finite-dimensional spaces, where impulse instants appear periodically. We present several characterizations on the stabilization; show how to design feedback laws; and provide locations for impulse instants to ensure the stabilization. In the proofs of these results, we set up a discrete LQ problem; derived a discrete dynamic programming principle, built up a variant of Riccati's equation; applied repeatedly the Kalman controllability decomposition; and used a controllability result built up in [17].
In this paper, we address an output feedback stabilization problem for a class of linear impulsive systems that accommodate arbitrarily-spaced impulse times and possibly singular state transition matrices. By combining recent results for state feedback stabilization and state estimation, we show that a separation property holds and formulate an output feedback compensation scheme in which the feedback loop is closed between a discrete-time measurement and a continuoustime control input. Rather than directly adopting an observerbased structure involving the time-varying gains associated with the separate stabilization and estimation problems, we construct a purely discrete-time compensator followed by a memoryless generalized hold device that achieves closed-loop exponential stability. This paper establishes the equivalence of three stabilizability-related properties for a class of linear impulsive systems. The first involves a gramian-based condition inspired by results for time-varying, discrete-time linear systems introduced decades ago. The second is the ability to achieve closed-loop exponential stability via state feedback. Finally, the third property is exponential stability of an ‘unreachable’ subsystem identified from a decomposition of the original system derived from an invariant subspace that characterizes the set of reachable states. A consequence of this analysis is that full state reachability of a linear impulsive system is not necessary for state feedback stabilization, a well-known fact for linear time-invariant systems. The main ideas of the paper are applied to the problem of synchronizing two Lorenz oscillators using underactuated impulsive control.