Geodesic
The Krylov space and the Kalman equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 1, pp. 5-10

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An optimal, in some respects, representation of vectors in the Krylov space is built with the help of a variational problem. The extremum of the variational problem is the solution of the Kalman matrix equation and the 2-norm of the solution is suggested to use as a characteristic of the Krylov space. This characteristic can also be used as the measure of controllability in stationary discreet problems of optimal control. 
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S. K. Godunov; V. M. Gordienko. The Krylov space and the Kalman equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 1, pp. 5-10. http://geodesic.mathdoc.fr/item/SJVM_1998_1_1_a1/