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].
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.