Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. In the period from 1967 till 1970 Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. In the present paper our aim is to bring up this calculus to the attention of researchers and demonstrate its usefulness.

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The main results in theory of controllability are formulated for deterministic or stochastic control systems given in a standard form. i.e., given as a first order differential equation driven by an infinitesimal generator of strongly continuous semigroup in an abstract Hilbert or Banach space and disturbed by a deterministic function or by a white noise process. At the same time, some deterministic or stochastic linear systems can be written in a standard form if the state space is enlarged. Respectively, the ordinary controllability conditions for them are too strong since they assume extended state space. It is reasonable to introduce partial controllability concepts, which assume original state space. In this paper, we study necessary and sufficient conditions of partial controllability for deterministic and stochastic linear control systems given in a standard form and their implications to particular cases.