TSUJIOKA, K: Remarks on Controllability of Euler-Bernoulli Equations with a Singular Boundary Condition
Saitama Mathematical Journal 1997(Vol. 15) pp.73-83
We shall prove approximate controllability for a system of the
Euler-Bernoulli equation (resp. damped wave equation)
utt(x,t)+2butxxxx(x,t)+
auxxxx(x,t) = xf(t)
(resp. utt(x,t)-2butxx(x,t)-auxx(x,t) = xf(t))
u(x,0) = u0 (x),
ut(x,0) = u1(x)
u(0,1) = ux(0,t) =
uxx(1,t) = 0
(resp. u(0,t) = 0)
uxxx(1,t) + kuxxxx(1,t) = 0
(resp. ux(1,t) + kuxx(1,t)
= 0)
(x in [0, 1], t in [0, T])
where a, b, k and T are positive constants and
f = f(t) is a control function.
The last boundary condition is singular in the sense
that its order 4 (resp 2) in x-derivative is same as that of PDE.
To remove the singularity we put
U(x, t)=(u(x, t), u(1, t)) in
H = L2(0,1) X R and obtain a second
order evolution equation in H which can be solved by Hille-Yosida
semigroup theory.