On a linear control problem under interference

This paper considers a linear control problem under the action of uncontrolled interference. The control process occurs in a given time interval. The possible values of interference belong to a compact set. The control is sought as the product of a scalar function and a vector function. Values of the vector function belong to a connected symmetric compact. This definition of control arises in control problems for mechanical systems of variable composition. For example, the law of variation of a reaction mass is defined as a function of time, and the control affects the direction of relative velocity in which the mass is separated. The terminal part of the board depends on the modulus of a linear function of the state vector. The integral part of the board is an integral over the interval of a degree of the scalar function. The control problem is considered within the theory of guaranteed result optimization. With the help of a linear change of variables, the control problem comes down to a one-type differential game. An optimal control existence theorem is proved under rather wide constraints on the class of problems. Necessary and sufficient conditions are found, under which an admissible control is optimal.