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Indefinite stochastic linear quadratic control and generalized differential Riccati equation. (English) Zbl 1009.93082
The paper deals with the indefinite LQ control problem in which the cost weighting matrices are indefinite in sign. This paper solves this open problem for LQ control in a finite time horizon. A new type of differential Riccati equation, called the generalized (differential) Riccati equation, is introduced, which involves algebraic equality/inequality constraints and a matrix pseudoinverse. It is then shown that the solvability of the generalized Riccati equation is not only sufficient, but also necessary, for the well-posedness of the indefinite LQ problem and the existence of optimal feedback/open-loop controls. Moreover, all of the optimal controls can be identified via the solution to the Riccati equation.
MSC:
93E20Optimal stochastic control (systems)
49K45Optimal stochastic control (optimality conditions)