Viscoelastic materials are widely used to control vibrations. However, their mechanical properties are known to be frequency and temperature-dependent. Thus, in a narrow frequency bandwidth, there is an optimal temperature that corresponds to a maximum loss factor and it is tricky to get a high damping level over a wide frequency range. Furthermore, an optimal temperature for a maximum structural damping leads to a low static stiffness because the peak of the loss factor is obtained during the glass transition when the storage modulus is decreasing. In order to obtain a compromise between stiffness and damping it is suggested to use a viscoelastic material which properties are functionally graded thanks to a non-uniform temperature field over the structure. In this work, a composite structure has been designed integrating a viscoelastic core and a heat control device. The optimal temperature field has been obtained through the minimization of a cost function that reflects the compromise between structural damping over a wide frequency band and high static rigidity. The experimental validation has been performed on a reduced scale airplane model: the composite wings are sandwich structures made of aluminum skins and a viscoelastic core in tBA/PEGDMA with a non-uniform temperature field and skins are in an aluminum and FR-4. A broadband excitation is produced with a shaker and the measurements are performed with a set of accelerometers. Several temperature fields are tested. The frequency response functions show the compromise obtained between static and dynamic behaviors when using the optimal temperature field determined by numerical simulation.