Distribution of terminal cost functional in continuous-time controlled system with noise-corrupted state information

In order to obtain the cumulative distribution function of the terminal cost in a scalar linear continuous-time system with noise corrupted measurements a first-order partial differential equation is derived. The system is subject to a linear saturated control strategy. The probability density function of the initial state and the history of the probability density function of the estimator error are assumed to be known. An illustrative numerical example is presented.