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 poor static stiffness because the peak of the loss factor is obtained during the glass transition when the storage modulus is decreasing. Additionally, in industrial applications, the requirements might change according to the system life-cycle. For instance, the stabilization functions that are used for optronics applications require high stiffness for positioning steps, and high damping for filtering functions. To achieve this goal, engineers usually use several viscoelastic materials with functionally graded damping properties. This allows obtaining a high loss factor over a wide frequency range. This solution is however not adaptive. In order to be able to adjust the properties in real time, we suggest in this paper to use a single material which properties are functionally graded thanks to a non-homogeneous temperature field over the structure. A composite structure has been numerically designed integrating a viscoelastic core and a heat control device. The optimal temperature field has been obtained based on the static and dynamic elastic strain energy densities that reflects the compromise between structural damping over a wide frequency band and high static rigidity.