**4. Feasibility Property**

Based on a qualitative analysis [24] of the solutions of systems of differential equations, the feasibility means that small changes in the model do not lead to a loss of quality. In other words, it is necessary that the solution has the contraction property.

**Hypothesis 1.** *A mathematical model is feasible, if its errors do not increase in time.*

**Definition 1.** *The system of differential equations is practically feasible, if this system as a oneparametric mapping obtains a contraction property in the implementation domain.*

Consider a system of differential equations

$$
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}),
\tag{24}
$$

where **<sup>x</sup>** <sup>∈</sup> <sup>R</sup>*n*.

Any ordinary differential equation is a recurrent description of a time function. A solution of the differential equation is a transformation from a recurrent form to a usual time function.

Computer calculation of the differential Equation (24) has a form

$$\mathbf{x}(t + \Delta t) = \mathbf{x}(t) + \Delta t \mathbf{f}(\mathbf{x}(t)),\tag{25}$$

where *t* is an independent parameter, Δ*t* is a constant parameter, and it is called a step of integration.

The right side of the Equation (25) is a one-parametric mapping from space R*<sup>n</sup>* to itself

$$F(\mathbf{x}, t) = \mathbf{x}(t) + \Delta t \mathbf{f}(\mathbf{x}(t)) : \mathbb{R}^n \to \mathbb{R}^n. \tag{26}$$

Let a compact domain D be set in the space R*n*. All solutions of the differential Equations (24), that are of our interest, belong to this domain. Therefore, for the differential Equations (24) the initial and terminal conditions belong to this domain

$$\mathbf{x}(0) \in \mathcal{D} \subseteq \mathbb{R}^n, \; \mathbf{x}(t\_f) \in \mathcal{D} \subseteq \mathbb{R}^n,\tag{27}$$

where **x**(*tf*) is a terminal point of the solution (24).

**Theorem 2.** *In domain* D *for the mapping (26), the following property is performed*

$$
\rho(\mathbf{x}^a(t), \mathbf{x}^b(t)) \le \rho(F(\mathbf{x}^a(t), t), F(\mathbf{x}^b(t), t)), \tag{28}
$$

*where* **<sup>x</sup>***a*(*t*) <sup>∈</sup> <sup>D</sup>*,* **<sup>x</sup>***b*(*t*) <sup>∈</sup> <sup>D</sup>*, <sup>ρ</sup>*(**x***a*, **<sup>x</sup>***b*) *is a distance between two points in the space* <sup>R</sup>*<sup>n</sup>*

$$\rho(\mathbf{x}^a, \mathbf{x}^b) = \left\| \mathbf{x}^a - \mathbf{x}^b \right\|. \tag{29}$$

*Then the mathematical model (24) is feasible if the domain* <sup>D</sup> <sup>⊆</sup> <sup>R</sup>*<sup>n</sup> according to the hypothesis.*

**Proof.** Let **x**(*t*) ∈ D be a known state of the system in the moment *t* and **y**(*t*) ∈ D be a real state of the system in the same moment. The error of the state is

$$\boldsymbol{\delta}(t) = \rho(\mathbf{x}(t), \mathbf{y}(t)). \tag{30}$$

According to the mapping (26)

$$\delta(t + \Delta t) = \rho(F(\mathbf{x}(t), t), F(\mathbf{y}(t))).\tag{31}$$

And according to the condition (28) of the theorem

$$
\delta(t) \le \delta(t + \Delta t). \tag{32}
$$

This proves the theorem.

The condition (28) shows that the system of differential equations as a one-parametric mapping has contraction property.

Assume that the system (24) in the neighborhood of the domain D has one stable equilibrium point, and there is no other equilibrium point in this neighborhood

$$\mathbf{f}(\bar{\mathbf{x}}) = \mathbf{0},\tag{33}$$

$$\det(\lambda \mathbf{E} - \mathbf{A}(\bar{\mathbf{x}})) = \lambda^n + a\_{n-1}\lambda^{n-1} + \dots + a\_1\lambda + a\_0 = \prod\_{j=1}^n (\lambda - \lambda\_j) = 0,\tag{34}$$

where **E** is a unit *n* × *n* matrix,

$$\mathbf{A}(\ddot{\mathbf{x}}) = \frac{\partial \ddot{\mathbf{f}}(\mathbf{x})}{\partial \mathbf{x}},\tag{35}$$

$$
\lambda\_j = \alpha\_j + i\beta\_{j\prime} \tag{36}
$$

*<sup>α</sup><sup>j</sup>* <sup>&</sup>lt; 0,*<sup>i</sup>* <sup>=</sup> √−1, *<sup>j</sup>* <sup>=</sup> 1, . . . , *<sup>n</sup>*.

**Theorem 3.** *If for the system (24) there is a domain* D *that includes one stable equilibrium point (33)–(36), then the system (24) is practically feasible.*

**Proof.** According to the Lyapunov's stability theorem on the first approximation the trivial solution of the differential Equation (24)

$$\mathbf{x}(t) = \bar{\mathbf{x}} = \text{constant} \tag{37}$$

is stable. This means, that, if any solution begins from other initial point **<sup>x</sup>**<sup>0</sup> <sup>=</sup> **<sup>x</sup>**˜, then it will be approximated to the stable solution asymptotically

$$
\rho(\mathbf{x}(t+\Delta t, \mathbf{x}^a), \bar{\mathbf{x}}) \le \rho(\mathbf{x}(t, \mathbf{x}^a), \bar{\mathbf{x}}), \tag{38}
$$

where **x**(*t*, **x***a*) is a solution of the differential Equation (24) from initial point **x***a*.

The same is true for another initial condition **x***<sup>b</sup>*

$$
\rho(\mathbf{x}(t+\Delta t, \mathbf{x}^b), \tilde{\mathbf{x}}) \le \rho(\mathbf{x}(t, \mathbf{x}^b), \tilde{\mathbf{x}}).\tag{39}
$$

From here, it follows that the domain *D* has a fixed point **x**˜ of contraction mapping [24], therefore distance between solutions **x**(*t*, **x***a*) and **x**(*t*, **x***b*) also tends to zero or

$$
\rho(\mathbf{x}(t+\Delta t, \mathbf{x}^a), \mathbf{x}(t+\Delta t, \mathbf{x}^b)) \le \rho(\mathbf{x}(t, \mathbf{x}^a), \mathbf{x}(t, \mathbf{x}^b)).\tag{40}
$$

This proves the theorem.

Following the principle of feasibility, an approach is proposed in which the optimal control problem is solved after ensuring the stability of the object in the state space. This approach is called the method of synthesized optimal control. It includes two stages. In the first stage, the system without perturbations is made stable in some point of the state space. This stage of synthesis of the stabilization system allows to embed the control in the object so that the system of differential equations would have the necessary property of feasibility. In this case, the equilibrium point can be changed after some time, but the object maintains equilibrium at every moment in time. Then we control the position of the stable equilibrium point, as an attractor, to solve the optimal control problem.
