Exact Controllability of the Euler–Bernoulli Equation with Controls in the Dirichlet and Neumann Boundary Conditions: A Nonconservative Case

Abstract

This paper considers the Euler–Bernoulli problem (1.1a–d) with boundary controls $g_1 $, $g_2 $ in the Dirichiet and Neumann boundary conditions, respectively. Several exact controllability results are shown, including the following. Problem (1.1a–d) is exactly controllable in an arbitrarily short time $T > 0$ in the space (of maximal regularity) $H^{ - 1} (\Omega ) \times V'$, V as in (1.4), (i) with boundary controls $g_1 \in L^2 (\Sigma )$, $g_2 \equiv 0$ under some geometrical conditions on $\Omega $; (ii) with boundary controls $g_1 \in L^2 (\Sigma )$ and $g_2 \in L^2 (0,T;H^{ - 1} (\Gamma ))$ without geometrical conditions on $\Omega $. A direct approach is given, based on an operator model for problem (1.la–d) and on multiplier techniques. An additional difficulty of the particular boundary conditions is due to the fact that, in the natural norms for the solution arising in the application of multiplier techniques, the corresponding homogeneous problem is not conservative. This difficulty is overco...