Replacing methane by hydrogen in premixed burners is a typical decarbonation scenario. This is usually done by ensuring a constant flame speed, i.e. by using ultra-lean H2 mixtures. This change can modify the thermoacoustic properties of burners. This study focuses on one class of thermoacoustic instabilities, Intrinsic ThermoAcoustic (ITA) modes, for flames stabilized on a diaphragm. An acoustic network approach is built to construct an ITA stability criterion (called ITAS criteria) for a three-duct configuration and a fully analytical expression is obtained in the limit of thin diaphragms. It shows that instability will grow when the Flame Transfer Function (FTF) gain n−π, located at the frequency at which u′-q′ phase ϕ=−π, is greater than a critical gain nc expressed by the ITAS criteria. Two-dimensional DNS, with anechoic terminations, of methane–air and hydrogen–air flames stabilized on a diaphragm are simulated with similar laminar flame speed sL0 and theoretical flame length, with an equivalence ratio of 0.73 and 0.4 respectively. The theory developed allows to analyze the differences with these two strong unstable ITA modes. For ultra-lean hydrogen flame, preferential diffusion near the diaphragm lips and tip opening reduces the flame time delay and shifts the ITA frequency towards higher values compared to methane. The assessment of two new control strategies is possible with the help of the analytical network model by increasing nc. A first strategy involves the use of preheated leaner mixtures: increasing Tu and reducing ϕ while keeping the flame speed sL0 constant. A second strategy consists of modifying the combustor geometry by changing the section ratio S2/S0, between plenum and combustion chamber. Both strategies successfully mitigate the hydrogen–air ITA modes and seems promising strategies for methane cases despite the fact that instabilities are not fully mitigated for this fuel at the conditions of interest in this study.