Searches for counterparts of gravitational waves at very high energies with H.E.S.S

The recent milestones in multi-messenger astronomy have opened new ways to study the Unverse. The unprecedented gravitational wave (GW) follow-up campaigns established the power that the combination of different messengers has to identify and study the nature and evolution of astrophysical phenomena. Here we focus on the search for high-energy gamma ray emission as electromagnetic counterpart of compact binary coalescences with the H.E.S.S. Imaging Air Cherenkov Telescopes (IACTs). In this contribution, the optimized strategies developed specifically for the prompt follow-up of gravitational wave events with H.E.S.S are presented. As illustration, the successful observation campaigns up to this time will be described, including the ones during Observation Run O2 on the binary black hole (BH-BH) merger GW170814 and the binary neutron star (NS-NS) merger GW170817, and an update on recent events occurring during O3. Results of these searches are presented and the constraints that prompt observations can put on very-high-energy, non-thermal emission, are briefly discussed. Finally, an outlook on further improvements for the gravitational waves follow-up program with H.E.S.S. will be provided.