We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function and prove theorems on the closeness of the corresponding trajectories and the values of quality criteria.The theory of set-valued mappings has been extensively developed since the late 1960s. In [1], Hukuhara introduced the notions of derivatives and integrals of set-valued mappings and studied the relationship between them. Later, differential equations with Hukuhara derivatives were considered in [2], various definitions of solutions were given and theorems on the existence of these solutions were proved in [3], and the applicability of certain averaging schemes to them was considered in [4,5].Equations with Hukuhara derivatives were used for the investigation of certain properties of an "integral funnel" of a differential inclusion in a Banach space in [6] and for the investigation of equations with fuzzy initial conditions in [7,8].In the present paper, we consider the problem of control over a process described by a linear differential equation with Hukuhara derivative and terminal quality criterion. This problem is significantly simplified if the control function is approximated by a piecewise-constant function. We present an algorithm for the construction of this approximate piecewise-constant control and prove the closeness of the corresponding trajectories and the values of quality criteria.Let Conv( ) R n be the space of nonempty compact and convex subsets of the Euclidean space R n with Hausdorff metric h(⋅ ⋅), . Consider the following control system described by a linear differential equation with Hukuhara derivative:n is a set-valued mapping that determines the state of the system, D X t h ( ) is the Hukuhara derivative [1], A t ( ) is an n n × matrix, F T (⋅) [ ] :, 0 → Conv( ) R n is the deviation of the system, and u(⋅) ∈ U ∈ Conv( ) R n is a control action.

In this article we prove that for any measurable admissible control ( ) w ⋅ and for any 0 ε > there exists piecewise constant admissible control ( ) w ⋅ such that for set solutions of control set system are ε -neighbouring.

In this article we prove the substantiation of the method of full averaging for the set integrodifferential equations with small parameter.