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Discrete-time indefinite LQ control with state and control dependent noises. (English) Zbl 1035.49024
Summary: This paper deals with the discrete-time stochastic LQ problem involving state and control dependent noises, whereas the weighting matrices in the cost function are allowed to be indefinite. In this general setting, it is shown that the well-posedness and the attainability of the LQ problem are equivalent. Moreover, a generalized difference Riccati equation is introduced and it is proved that its solvability is necessary and sufficient for the existence of an optimal control which can be either of state feedback or open-loop form. Furthermore, the set of all optimal controls is identified in terms of the solution to the proposed difference Riccati equation.
MSC:
49N10Linear-quadratic optimal control problems
93C55Discrete-time control systems
49J55Optimal stochastic control (existence)
93E20Optimal stochastic control (systems)