Noboru Kunimatsu

Modified Gronwall's inequality and its application to stabilization problem for semilinear parabolic systems

Abstract This paper deals with the stabilization of semilinear parabolic distributed systems by means of boundary controls. The partial differential equation describing the system is formulated as a semilinear evolution equation in a Hilbert space by using a variable transformation. Here it is shown that the solution globally exists for the closed-loop system which installs a finite-dimensional dynamic compensator by using the properties of an analytic semigroup. Moreover, a sufficient condition for stabilization of the system is derived by making use of the modified Gronwalls inequality developed in this paper.

Semigroup model and stability of the structurally damped Timoshenko beam with boundary inputs

Abstract This paper is concerned with the linear partial differential equation describing the motion of the Timoshenko beam with viscous internal dampings and with a certain type of boundary inputs. The equation is formulated as an evolution equation of the form dz( t)/dt = — Az( i) in a Hilbert space, and then it is proved for the linear operator — A to generate an analytic semigroup. Moreover, the stability of the boundary control system is examined. It is shown that in some cases the present type of boundary inputs degrades the degree of stability of the system. This is a remarkable contrast to the results of previous research which states that the same type of boundary inputs are effctive in stabilizing the Timoshenko beam without any structural damping.

Stabilization of a non-linear distributed parameter vibratory system

The feedback stabilization problem for a non-linear elastic distributed parameter system including a destabilizing effect is studied. The model is formulated as an evolution equation in a Hilbert space. Semigroup theory is used to investigate an initial value problem for the equation. Moreover, a control system with a finite dimensional controller is constructed. It is then proved that the state of the overall system converges to zero with a desirable exponential decay rate as time tends to infinity.

On the stability of a linear bioprocess model with recycle loop

In this note, we treat a linear bioprocess model with recycle loop and analyze the spectrum of the closed-loop operator in order to discuss the exponential stability. It is shown that a C/sub 0/-semigroup generated by the closed-loop operator satisfies the spectrum determined growth assumption and that the recycle loop works effectively for the process through a numerical simulation.

Compensator design of semi-linear parabolic systems

The stabilization of semi-linear parabolic distributed systems by means of boundary controls is treated, formulating the partial differential equation describing the system as an evolution equation in a Hilbert space. Here, by using the properties of an analytic semigroup, the global existence of the solution under static output feedback is shown. Moreover, in the similar way, the global existence of the solution and the stabihzability for the system which installs a finite dimensional dynamic compensator is shown.

Feedback stabilization of infinite-dimensional systems with periodically time-varying input and output operators

The stabilization problem of infinite-dimensional systems with periodically time-varying bounded input and unbounded output operators is considered. The unboundedless of the periodically time-varying output operators is assumed to be Aγ -bounded. For example, the systems contain linear diffusion systems with boundary control and observation, where the actuator and sensor influence functions periodically vary with respect to the time variable. The aim of this paper is to show that a controller, which consists of a finite-dimensional stabilizing controller constructed for the finite-dimensional reduced-order modal model and a residual mode filter, works as a finite-dimensional stabilizing controller for the original infinite-dimensional system if the order of residual mode filter is chosen to be sufficiently large.