## Quadratic control problem for linear discrete-time time-varying systems with multiplicative noise in the Hilbert spaces.(English)Zbl 1097.93039

Summary: We prove that under stabilizability and uniform observability conditions the discretetime Riccati equation associated to the general quadratic control problem has a unique, uniformly positive, bounded on $$\mathbb N^*$$ solution. We use this result to solve the problem of existence of an optimal control for the cost function investigated.

### MSC:

 93E20 Optimal stochastic control 49N10 Linear-quadratic optimal control problems 39A11 Stability of difference equations (MSC2000)