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].
About general theory for impulse systems, we refer readers to @cite_6 @cite_22 @cite_23 and the references therein.
@cite_23 @cite_22 @cite_6
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Geared primarily to an audience consisting of mathematically advanced undergraduate or beginning graduate students, this text may additionally be used by engineering students interested in a rigorous, proof-oriented systems course that goes beyond the classical frequency-domain material and more applied courses. The minimal mathematical background required is a working knowledge of linear algebra and differential equations. The book covers what constitutes the common core of control theory and is unique in its emphasis on foundational aspects. While covering a wide range of topics written in a standard theorem proof style, it also develops the necessary techniques from scratch. In this second edition, new chapters and sections have been added, dealing with time optimal control of linear systems, variational and numerical approaches to nonlinear control, nonlinear controllability via Lie-algebraic methods, and controllability of recurrent nets and of linear systems with bounded controls. A new hyperchaotic system has been proposed recently. It is generated by controlling a unified chaotic system to hyperchaotic via a simple technique using a sinusoidal parameter perturbation control input. In this paper, we further investigate its dynamical behaviors, its circuit implementation and its impulsive control. Different chaotic attractors are illustrated by both numerical simulations and electronic experiments. It is also shown that the new hyperchaotic system can be stabilized by impulsive control. Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, do exhibit impulsive effects. Thus impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems.