We consider the problem of minimizing an integral functional on the solutions of a controlled system described by a non-linear differential equation in a separable Banach space and a variational inequality. The variational inequality determines a hysteresis operator whose input is a trajectory of the controlled system and whose output occurs in the right-hand side of the differential equation, in the constraint on the control, and in the functional to be minimized. The constraint on the control is a multivalued map with closed non-convex values and the integrand is a non-convex function of the control. Along with the original problem, we consider the problem of minimizing the integral functional with integrand convexified with respect to the control, on the solutions of the controlled system with convexified constraints on the control (the relaxed problem).

By a solution of the controlled system we mean a triple: the output of the hysteresis operator, the trajectory, and the control. We establish a relation between the minimization problem and the relaxed problem. This relation is an analogue of Bogolyubov’s classical theorem in the calculus of variations. We also study the relation between the solutions of the original controlled system and those of the system with convexified constraints on the control. This relation is usually referred to as relaxation. For a finite-dimensional space we prove the existence of an optimal solution in the relaxed optimization problem.

我们考虑在可分离的 Banach 空间和变分不等式中最小化由非线性微分方程描述的受控系统的解的积分泛函的问题。变分不等式确定了一个滞后算子，其输入是受控系统的轨迹，其输出出现在微分方程的右侧、控制约束中以及要最小化的泛函中。控件上的约束是具有封闭非凸值的多值映射，被积函数是控件的非凸函数。与原始问题一起，我们考虑最小化积分泛函的问题，被积函数相对于控制凸化，

通过受控系统的解决方案，我们指的是三元组：滞后算子的输出、轨迹和控制。我们建立了最小化问题和松弛问题之间的关系。这种关系类似于博戈柳博夫在变分法中的经典定理。我们还研究了原始受控系统的解与对控制具有凸化约束的系统的解之间的关系。这种关系通常称为松弛。对于有限维空间，我们证明了松弛优化问题中最优解的存在。