However, I am interested to know whether one could determine the stability of the shocks analytically.
Well if the shock wave is stationary, then it is stable, at least locally and for the time being. A good starting point is looking at the Rankine–Hugoniot relations. If those hold true, then the shock is stable and stationary. When additional sources of energy dissipation are necessary or additional conservation terms are necessary in the Rankine–Hugoniot relations, then the shock is not stable unless those extra terms balance the nonlinear steepening with energy dissipation.
As for references, e.g., see https://physics.stackexchange.com/a/139436/59023 and references therein. Another answer with some useful references is found at https://physics.stackexchange.com/a/281988/59023. An answer to a question about the "death" of a shock wave after the piston/driver stops is at https://physics.stackexchange.com/a/369511/59023.
There is also a great paper by Petschek  on "Aerodynamic Dissipation," which discusses a lot of the fundamentals of shock initiation and stability.
- Coroniti, F.V. "Dissipation discontinuities in hydromagnetic shock waves," J. Plasma Phys. 4, 265, doi:10.1017/S0022377800004992, (1970).
- Krasnoselskikh, V.V., B. Lembège, P. Savoini, and V.V. Lobzin "Nonstationarity of strong collisionless quasiperpendicular shocks: Theory and full particle numerical simulations," Phys. Plasmas 9, 1192-1209, doi:10.1063/1.1457465, (2002).
- Petschek, H.E. "Aerodynamic Dissipation," Rev. Modern Phys. 30(3), pp. 966-974, doi:10.1103/RevModPhys.30.966, 1958.
- Shukla, P.K., B. Eliasson, M. Marklund, and R. Bingham "Nonlinear model for magnetosonic shocklets in plasmas," Phys. Plasmas 11(5), pp. 2311-2313, doi:10.1063/1.1690297, (2004).
- Tidman, D.A., and T.G. Northrop "Emission of plasma waves by the Earth’s bow shock," J. Geophys. Res. 73, 1543–1553, doi:10.1029/JA073i005p01543, (1968).
- Whitham, G. B. (1999), Linear and Nonlinear Waves, New York, NY: John Wiley & Sons, Inc.; ISBN:0-471-35942-4.
- Zel'dovich, Ya.B., and Yu.P. Raizer (2002) Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Ed. by W.D. Hayes and R.F. Probstein, Mineola, NY, Dover Publications, inc., The Dover Edition; ISBN-13: 978-0486420028.