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 @cite_1 , the author built up a Kalman-type controllability decomposition for the system: @math Based on the decomposition, a necessary condition, as well as a sufficient condition, for the stabilization of the above system was given. Both results are related to some kind of reachability. The stabilization of the above system was also studied in @cite_12 .
@cite_1 @cite_12
2581030625 2074704816 2161442655 2757755218
Abstract This paper studies the approximate and null controllability for impulse controlled systems of heat equations coupled by a pair ( A , B ) of constant matrices. We present a necessary and sufficient condition for the approximate controllability, which is exactly Kalman's controllability rank condition of ( A , B ) . We prove that when such a system is approximately controllable, the approximate controllability over an interval [ 0 , T ] can be realized by adding controls at arbitrary q ( A , B ) different control instants 0 τ 1 τ 2 ⋯ τ q ( A , B ) T , provided that τ q ( A , B ) − τ 1 d A , where d A ≜ min ⁡ π | Im λ | : λ ∈ σ ( A ) and q ( A , B ) ≤ n . We also show that in general, such systems are not null controllable. This paper studies the controllability and observability of a class of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for observability and determinability of such systems are established, respectively. It is also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of geometric type, they can be transformed into algebraic type conveniently. Finally, a numerical example is given to illustrate the utility of our criteria. 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. This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the closed-loop system decays exponentially to zero in the @math --norm and the decay rate can be tuned to be as large as desired if the initial data is small enough.