In this paper, we consider the design of an H1 trade-off dependent controller, that is, a controller such that, for a given Linear Time-Invariant plant, a set of performance trade-offs parameterized by a scalar µ is satisfied. The controller state space matrices are explicit functions of µ. This new problem is a special case of the design of a parameter dependent controller for a parameter dependent plant, which has many application in Automatic Control. This last design problem can be naturally formulated as a convex but infinite dimensional optimization problem involving parameter dependent Linear Matrix Inequality (LMI) constraints. In this paper, we propose finite dimensional (parameter independent) LMI constraints which are equivalent to the parameter dependent LMI constraints. The parameter dependent controller design is then formulated as a convex finite dimensional LMI optimization problem. The obtained result is then applied to the trade-off dependent controller design. A numerical example emphasizes the strong interest of our finite dimensional optimization problem with respect to the trade-off dependent control application.