On stochastic optimality for a linear controller with attenuating disturbances

Stabilization of linear stochastic systems with a discount: Modeling and estimation of the long-term effects from the application of optimal control strategies

“…For bounded (i.e., in positive and zero discounting), the statement of Theorem 1 is a conse quence of the application of the result on the average optimality for systems with a time varying of param eters of the perturbation process, see [5]. Indeed, having changed variables we obtain the equation of the dynamics of the process +…”

“…and an arbitrarily small number Under the conditions of (a), by finding we obtain the exponential estimate of type (5). In (b) we assign…”

“…The objective functional in new variables = in this Then, on condition of the fulfillment of assumptions and according to the state ment of Theorem 1 from [5], control is average optimal over an infinite time horizon in the sense of the solution of problem The inverse change of variables to those made earlier, as well as assumption lead to the optimal control U* of type (6) and to its corresponding process (7) in the initial system with discounting. For the case of the negative discounting (increasing ), Theorem 1 was proved in [3, Theorem 3] with assumptions and Note 1.…”

“…We note that a decreasing corresponds to positive time preferences ( ), in the case of negative time preferences increases ( ), if then the time preferences are zero ( ). To estimate the performance of control actions chosen in order to stabilize the system in a long term period, consider the criterion of optimality proposed in [3,5]. We will try to find control such that the correlation (3) is fulfilled for any at a constant independent of control.…”

“…It is natural to assign the optimality functional in the integral quadratic form [4] and to include the discounting function there reflecting the subjects' time preferences. For the decisions to be made on unbounded planning hori zons, a criterion is used that includes the minimization of the upper limit from the expected losses per unit of the accumulated discount [3,5]. The thus defined law of control will be called the average optimal over an infinite time horizon.…”