## 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)
*u*_{tt}(*x,t*)+2*bu*_{txxxx}(*x,t*)+
*au*_{xxxx}(*x,t*) = *xf*(*t*)

(resp. *u*_{tt}(*x,t*)-2*bu*_{txx}(*x,t*)-*au*_{xx}(*x,t*) = *xf*(*t*))

*u*(*x*,0) = *u*_{0} (*x*),
*u*_{t}(*x*,0) = *u*_{1}(*x*)

*u*(0,1) = *u*_{x}(0,*t*) =
*u*_{xx}(1,*t*) = 0
(resp. *u*(0,*t*) = 0)

*u*_{xxx}(1,*t*) + *ku*_{xxxx}(1,*t*) = 0
(resp. *u*_{x}(1,*t*) + *ku*_{xx}(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* = *L*^{2}(0,1) X *R* and obtain a second
order evolution equation in *H* which can be solved by Hille-Yosida
semigroup theory.