This talk surveys theories of control in the presence of noise, their successes and limits. Throughout we pay particular attention to what types of noise models can be handled. In optimal stochastic control, one assumes a full statistical description of the noise and translates the design objective into a problem of minimizing an expected cost. The classical LQG problem so prominent in applications is an example. In H control, no model on the noise is assumed; it is considered to be an unknown, possibly random disturbance. The design objective is to protect against worst case disturbances by looking for feedback controls that give dissipative and stable systems. The important modelling constraint for success in stochastic optimal control is summarized in the word "Markov." In practice this means that the noise is independent increment, white noise, and that state feedback control leads to Markov processes. In the LQG case the analysis can be framed in an L2 setting in which only first and second moments of the noise are modelled. We explain this and relate it to the basis of the optimization theory, dynamic programming. We shall also summarize the H theory. Interestingly, the H theory also involves. We shall summarize the approach and describe its relation to risk sensitive stochastic control.
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