A variational approach to constrained controllability for distributed systems

We consider linear distributed control systems of the form y ′ ( t ) = A y ( t ) + B u ( t ) where A generates a strongly continuous semigroup ( e t A ) t ⩾ 0 on an infinite dimensional Hilbert space Y. We suppose that the control operator B is bounded from the (Hilbert) control space U to Y. Taking into account eventual control constraint (such as saturation), we study the problem of exact controllability by using a variational approach. Applications to hyperbolic-like systems of the form z ″ ( t ) + A z ( t ) = B u ( t ) are treated.