Two problems related to the optimization of linear stationary dynamic systems are considered. A general formulation of the multi-purpose problem of optimal control with the choice of design parameters is given. As a special case, the problem of multi-objective optimization of a linear system according to an integral quadratic criterion with a given random distribution of initial deviations is considered. The solution is based on the method of simultaneously reducing two positive-definite quadratic forms to diagonal form. Analytical results have been obtained that make it possible to calculate the mathematical expectation of the criterion under the normal multidimensional distribution law of the vector of random initial perturbations. The inverse problem of stability theory is formulated: to find a vector of structural parameters that ensure the stability of the system and a given average value of the quadratic integral quality criterion on a set of initial perturbations. The solution of the problem is proposed to be carried out in two stages.
