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_9 , the authors studied the controllability for the system: @math (Here @math , @math and @math .) They found @math (defined in ) with @math and @math ) so that for each @math and each @math with @math , the above system is controllable, provided that @math holds Kalman controllability rank condition. This result is used in the proofs of Theorem , as well as Theorem .
@cite_9
2581030625 2074704816 2126185507 2006705367
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. For the multivariable control system described by [ x = Ax + Bu, y = Cx, z = Dx, ] necessary and sufficient conditions are found for the existence of state feedback @math such that @math and @math as @math . It is assumed that @math is A-invariant, or equivalently that a dynamic observer is utilized. A constructive version of the existence conditions is obtained under the further assumption that auxiliary integrating elements can be introduced by way of dynamic compensation, and a bound is given for the number of such elements required. This paper presents a bounded model checking tool called @math for hybrid systems. It translates a reachability problem of a nonlinear hybrid system into a predicate logic formula involving arithmetic constraints and checks the satisfiability of the formula based on a satisfiability modulo theories method. We tightly integrate (i) an incremental SAT solver to enumerate the possible sets of constraints and (ii) an interval-based solver for hybrid constraint systems (HCSs) to solve the constraints described in the formulas. The HCS solver verifies the occurrence of a discrete change by using a set of boxes to enclose continuous states that may cause the discrete change. We utilize the existence property of a unique solution in the boxes computed by the HCS solver as (i) a proof of the reachability of a model and (ii) a guide in the over-approximation refinement procedure. Our @math implementation successfully handled several examples including those with nonlinear constraints.