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)+2bu_txxxx(x,t)+ au_xxxx(x,t) = xf(t)
(resp. u_tt(x,t)-2bu_txx(x,t)-au_xx(x,t) = xf(t))
u(x,0) = u₀ (x), u_t(x,0) = u₁(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 = L²(0,1) X R and obtain a second order evolution equation in H which can be solved by Hille-Yosida semigroup theory.