Stabilization and disturbance rejection for the wave equation. (English) Zbl 0907.93047

Morgül, Ömer

IEEE Trans. Autom. Control 43, No. 1, 89-95 (1998).

The paper studies the asymptotic stabilization of a wave equation in a bounded interval \((0,1) \subset \mathbb{R}^1\). The equation with state \(y(\cdot,t)\) is described by \[ y_{tt} -y_{xx} =0, \quad x\in(0,1),\;t\geq 0, \]
\[ y(0,t)=0, \quad y_x(1,t)= -f(t),\;t\geq 0, \]
\[ y(x,0) =w_0, \quad y_t (x,0)= w_1,\;x\in (0,1). \] Here, \(f(t)\) denotes a boundary input to be designed as a suitable feedback of the output: \((y(1,t), y_t(1,t))\), \(t\geq 0\). It is designed via a finite-dimensional compensator with state \((z_1,x_1,x_2)\) and parameters such as matrix \(A\), vectors \(b\) and \(c\), etc. Under several assumptions such as the controllability of \((A,b)\) and the observability of \((c,A)\) etc., the designed composite control system with state \((y,y_t, z_1,x_1, x_2)\) generates a \(C_0\) semigroup \(T(t)\) which is dissipative. By applying LaSalle’s invariance principle, it is shown that \(T(t)\) asymptotically converges to 0. When \(f(t)\) is influenced by a disturbance, it is also discussed how to attenuate the effect of the disturbance.

Reviewer: T.Nambu (Kobe)

MSC:

93D15	Stabilization of systems by feedback
93C20	Control/observation systems governed by partial differential equations
93C73	Perturbations in control/observation systems

Keywords:

boundary control; dissipativeness; asymptotic stabilization; wave equation; finite-dimensional compensator; LaSalle’s invariance principle