Control and stochastic

Grimble, M.J. and Johnson, M.A. (1988) Optimal Control and Stochastic Estimation Theory and Application, Vols 1 and 2, John Wiley Sons, Chichester, UK. 

Figure 4. Comparison of operator control and stochastic control of viscosity on two parallel...

Van Slyke, R., and Wets, R. J. B. (1969), L-Shaped Linear Programs with Application to Optimal Control and Stochastic Programming, SIAM Journal of Applied Mathematics, Vol. 17, pp. 638-663. 

If there is no time-dependent external force, the dynamics of a molecular system will evolve on a constant-energy surface. Therefore, a natural choice of the statistical ensemble in molecular dynamics simulation is the micro-canonical ensemble (NVE). Other types of ensembles, such as the canonical ensemble (NVT) and the isothermal-isobaric ensemble (NPT), can also be realized by controlling corresponding thermodynamic variables. For the last two ensembles, the temperature of the ensemble needs to be controlled and four different control mechanisms, namely differential control, proportional control, integral control and stochastic control, have been developed in the literature. As an example, a proportional thermostat for the NVT ensemble will be briefly discussed as follows. 

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