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].
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. The present note investigates the stabilization, controllability and optimal control problem of Boolean networks with impulsive effects and state constraints. By using the semi-tensor product method, the algebraic form of the Boolean networks with impulsive effects and state constraints is derived. The stabilization and controllability issues of the systems are investigated and some necessary and sufficient conditions are obtained. In addition, the Mayer-type optimal control problem is also studied and algorithms are provided to design the control sequence. Furthermore, examples are given to illustrate the main results. 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. Many dynamic systems in physics, chemistry, biology, engineering, and information science have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes. These complex dynamic behaviors can be modeled by impulsive differential systems. This paper studies the controllability and observability for a class of time-varying impulsive control systems. Several sufficient and necessary conditions for state controllability and state observability of such systems are established and the corresponding criteria for time-invariant impulsive control systems are also obtained. Meanwhile, several new results associated with variation of parameters for time-varying impulsive control systems are derived. This paper is concerned with the controllability and observability for a class of piecewise linear time-varying impulsive systems. Several sufficient and necessary conditions for state controllability and observability of such systems are established. Meanwhile, corresponding criteria for time-invariant impulsive systems are also obtained and the criteria are compared with the existing results. For a linear impulsive system, the set of states that are reachable from the origin when the initial time, impulse times, and final time are fixed is contained in an invariant subspace determined by the system data. It is known that reversibility of the system is sufficient to yield, for a specified initial time, the existence of some impulse time set and final time for which the reachable set equals the invariant subspace. In this paper, we relax the reversibility requirement and present a condition that is necessary as well as sufficient under which this property holds. This new condition involves the property of achieving reversibility via feedback and admits an explicit geometric characterization. Moreover, this feedback-reversibility property only needs to hold for the subsystem defined as the full system restricted to the invariant subspace. We further show that feedback-reversibility of the restricted system ensures that the reachable set equals the invariant subspace for almost any impulse time set and final time for which the number of impulse times contained in the underlying time interval exceeds a lower bound. Many practical systems in physics, chemistry, biology, engineering, and information science have impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. These complex dynamical behaviors can be modeled by impulsive differential systems. This paper studies the controllability and observability issues for a general time-varying impulsive control systems. Sufficient and necessary conditions for state controllability and state observability of the impulsive control systems are established and their applications to time-invariant impulsive control systems are also discussed. Furthermore, several new results associated with variation of parameters for time-varying impulsive control systems are derived.