**3. Theoretical Background and Justifications for the Synthesized Optimal Control Method**

Problems with uncertainties are often considered in optimal control, since the question is relevant in the practical implementation of obtained systems. As a rule, uncertain parameters of the right-hand sides or initial conditions are considered as uncertainties, or some random perturbations are introduced. The main direction of solving problems with perturbations is to ensure the stability of the obtained solution. So, firstly, the problem of optimal control is solved without uncertainties, and then, using the stabilization system, an attempt is made to ensure the stability of motion relative to the optimal trajectory. In fact, the creation of a stabilization system is an attempt to ensure the stability of the differential equation solution according to Lyapunov.

**Theorem 1.** *To perform the condition (10) it is enough that a partial solution (8) of the system (9) without perturbations* **y**(*t*) ≡ 0

$$
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{h}(\mathbf{x}, t)),
\tag{18}
$$

*was stable according to Lyapunov.*

**Proof.** From differential Equation (1) follows

$$\mathbf{x}(t + \Delta t) = \mathbf{x}(t) + \Delta t \mathbf{f}(\mathbf{x}, \mathbf{h}(\mathbf{x}, t)) + \Delta t \mathbf{y}(t), \tag{19}$$

or

$$\bar{\mathbf{x}}(t) = \mathbf{x}(t, \mathbf{x}^0) + \mathbf{v}(t), \tag{20}$$

where

$$\mathbf{v}(t) = \int\_0^t \mathbf{y}(t)dt. \tag{21}$$

Let Δ*<sup>y</sup>* be given. Then according to condition (15) you can always define Δ˜ and value ¯ *δ* for perturbed solution **x**¯ such that according to condition of stability on Lyapunov [20,21]

$$\|\|\mathbf{x}(t, \mathbf{x}^0) - \bar{\mathbf{x}}(t)\|\| < \bar{\delta}, \ \forall t \in [0; t\_f]. \tag{22}$$

For this it is enough to satisfy the inequality

$$0 \le \|\mathbf{v}(t)\| \le \bar{\delta}/2, \ \forall t \in [0; t\_f]. \tag{23}$$

However, to find control function (7) such that partial solution (8) was stable according to Lyapunov is rather difficult and, in fact, it is not always necessary. According to Lyapunov's theorem, a stable solution to a differential equation must have the property of an attractor [20,22], and, therefore, from the mathematical point of view the synthesis of stabilization system is an attempt to give an attractor property to the found optimal trajectory [21,23]. The main problem of unstable solutions is that they are difficult to implement, since small perturbations of the model lead to large errors of the functional, in other words, the solution does not have the attractor property. But in fact, the requirement for the optimal solution to obtain the attractor property or be Lyapunov stable is a fairly strict one and it could be redundant, and other weaker requirements may be enough to implement the resulting solution. For example, the motion of a pendulum is not Lyapunov stable if it is not the zero rest point, but it is physically feasible, since its small perturbations lead to small perturbations of the functional.

In this concern let us introduce the concept of feasibility.
