To the formalization of the estimates of the phenomenon state
The questions of the formalization of the estimates of the phenomenon state, which can be used in the decision-making multicriterion problems [Vector-Valued Optimization Problems in Control Theory (Academic Press, New York, 1979)] taking into account the metrics of the space of partial criteria are considered. Any phenomenon is described by some system of characteristic parameters, which values are presented by the coordinates of points in the space of states of the given phenomenon. A notion of the "ideal" state determined by optimal values of each characteristic parameter separately and independently of others is used. Estimate of any state is determined by the distance between that state and the "ideal" one. That permits to choose from a multitude of the given states the best one, to which conforms a minimum distance to the "ideal" state. In the general case of the curved space the distances are measured by geodesic lines drawn by using a metric tensor of the given space. The metric tensor components themselves are determined by the solution of the differential equations set up from the condition of minimization of a certain functional.
metrics distance geodesic tensor curvature invariance