Abstract
In this paper, we consider the time-optimal control problem about a kind of Petrowsky system and its bang-bang property. To solve this problem, we first construct another control problem, whose null controllability is equivalent to the controllability of the time-optimal control problem of the Petrowsky system, and give the necessary condition for the null controllability. Then we show the existence of time-optimal control of the Petrowsky system through minimum sequences, for the null controllability of the constructed control problem is equivalent to the controllability of the time-optimal control of the Petrowsky system. At last, with the null controllability, we obtain the bang-bang property of the time-optimal control of the Petrowsky system by contradiction, moreover, we know the time-optimal control acts on one subset of the boundary of the vibration system.
Highlights
In physics, the Petrowsky systems are usually used to describe the phenomena about the vibration of elastic beams or perches, etc
We investigate the time-optimal control problem of a kind of Petrowsky system about the vibration of an elastic beam which is a joint of two parts of a building, where the elastic beam is used to reduce the damage caused by vibration to the building
With the null-controllability, we prove the bang-bang property of the time-optimal control of the Petrowsky system by contraction, and find out the time-optimal control just act on the boundary of the elastic beam
Summary
The Petrowsky systems are usually used to describe the phenomena about the vibration of elastic beams or perches, etc. We often need to control a certain vibration system with an external force to keep it in a given equilibrium state, where we may face a new problem: how to control the oscillation system to reach the equilibrium state as soon as possible This is a time-optimal control problem for vibration systems. The engineers usually control the damage of vibration by building some elastic beams, where the vibration of the elastic beam can be controlled under a target statement through a bounded outside force. With the null-controllability, we prove the bang-bang property of the time-optimal control of the Petrowsky system by contraction, and find out the time-optimal control just act on the boundary of the elastic beam
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