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The feedback control of a class of linear systems disturbed by additive white noise is considered. A linear control law is used to minimize a non-quadratic integral performance criterion. The stochastic problem is transformed to a deterministic problem, and a solution is developed using the calculus of variations. IntroductionOptimal feedback control policies for linear stochastic systems with quadratic performance indices are well known (Sage 1968, Meditch 1969. Solutions have been developed both for systems which have perfect observations of the state vector available, and for those with only noisy observations available. It is of interest to consider systems with non-quadratic cost functions, since in some instances, such performance measures may be a more realistic indicator of the design Objectives than the usual quadratic index.An unfortunate consequence of considering generalized performance measures, even for deterministic control problems, is that one must usually restrict the class of systems for which the design technique is applicable. Even under such restrictions explicit closed-loop control laws are rarely obtained, and the inherent computational difficulties involved in solving the two-point boundary value problems associated with open-loop controllers are present.In this paper a class of asymmetric linear stochastic systems is considered. In addition to the control input, there is a white noise input disturbance. It is assumed that the state vector is available to control the system, and a linear feedback control law is to be determined which minimizes an integral performance criterion.Deterministic versions of the problem considered here have been recently investigated . It has been shown that either a non-linear, inner product feedback control, or a linear time-variable feedback control were equivalent, in the sense that both minimized the designated cost function. Such an equivalence is not determined in this paper since the application of a non-linear feedback controller to a stochastic system makes analysis extremely difficult. Calculation of the moments for a non-linear stochastic system (Cumming 1967), in general, leads one to consider an infinite number of differential equations. It has been common in stochastic control theory to constrain the control structure to be linear prior to optimization (Axsater 1966, Sims andMelsa 1970). In this study, the linearity of the control structure enables one to proceed rather easily in analysis. Consideration of a simple