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.