Discrete Time
In a discrete time context, the decision-maker observes the state variable, possibly with observational noise, in each time period. The objective may be to optimize the sum of expected values of a nonlinear (possibly quadratic) objective function over all the time periods from the present to the final period of concern, or to optimize the value of the objective function as of the final period only. At each time period new observations are made, and the control variables are to be adjusted optimally. Finding the optimal solution for the present time may involve iterating a matrix Riccati equation backwards in time from the last period to the present period.
In the discrete-time case with uncertainty about the parameter values in the transition matrix (giving the effect of current values of the state variables on their own evolution) and/or the control response matrix of the state equation, but still with a linear state equation and quadratic objective function, a Riccati equation can still be obtained for iterating backward to each period's solution even though certainty equivalence does not apply.ch.13 The discrete-time case of a non-quadratic loss function but only additive disturbances can also be handled, albeit with more complications.
Read more about this topic: Stochastic Control
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