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The goal of this article is to discuss controllability properties for anabstract linear system of the form $y' = Ay + Bu$ under some additional linearprojection constraints on the control $u$ or / and on the controlled trajectory$y$. In particular, we discuss the possibility of imposing the linearprojections of the controlled trajectory and of the control, in the context ofapproximate controllability, exact controllability and null-controllability. Asit turns out, in all these settings, for being able to impose linear projectionconstraints on the trajectory and on the control, we will strongly rely on aunique continuation property for the adjoint system which, to our knowledge,has not been identified so far, and which does not seem classical. We shalltherefore provide several instances in which this unique continuation propertycan be checked.

Control issues and linear projection constraints on the control and on the controlled trajectory