**8. Programmed control with Prob. 1 for stochastic dynamical systems**

Definition 1.6 [18, 19]. A PCP1 is called a control of stochastic system which allows the preservation with probability 1 of a constant value for the same function which depends on this systems position for time periods of any length *T*.

Let us consider the stochastic nonlinear jump of diffusion equations system:

$$\begin{split}dX(t) &= (P(t, X(t)) + Z(t, X(t)) \cdot u(t, X(t)))dt + (B(t, X(t)) + K(t, X(t)))dw(t) + \\ &\quad + \int\_{R\_{\gamma}} (L(t, X(t), \gamma) + \Lambda(t, X(t))) \nu(dt, d\gamma), \end{split} \tag{24}$$

*Invariants for a Dynamical System with Strong Random Perturbations DOI: http://dx.doi.org/10.5772/intechopen.96235*

where *P*ð Þ� , *Z*ð Þ� are given matrix functions and *B*ð Þ� , *L*ð Þ� are the functions that may either be known or not. For such systems we construct a unit of programmed control f g *u t*ð Þ , *X t*ð Þ , *K t*ð Þ , *X t*ð Þ , *M t*ð Þ , *X t*ð Þ which allows the system (24) to be on the given manifold f*u t*ð ,*X t*ð ÞÞg ¼ f g *u*ð Þ 0, *x*<sup>0</sup> with Prob. 1 (PCP1) for each *t* ∈½ � 0, *T* , *T* ≤ ∞.

Suppose that the nonrandom function *s t*ð Þ , *X t*ð Þ is the first integral for the same stochastic dynamical system. The PCP1 f g *u t*ð Þ ,*X t*ð Þ ,*K t*ð Þ , *X t*ð Þ , *M t*ð Þ , *X t*ð Þ is the solution for the algebraic system of linear equations.

Theorem 1.4 Let a controlled dynamical system be subjected to Brownian perturbations and Poisson jumps. The unit of PCP1 f g *u t*ð Þ , *X t*ð Þ ,*K t*ð Þ , *X t*ð Þ , *M t*ð Þ , *X t*ð Þ , allowing this system to remain with probability 1 on the dynamically structured integral mfd *s t*ð Þ¼ ,*X t*ð Þ , *xo* ,*ω s*ð Þ 0, *xo* , is a solution of the linear equations system (with respect to functions **u**ð Þ *t*, **x**ð Þ*t* ), *K t*ð Þ ,*X t*ð Þ , *M t*ð Þ , *X t*ð Þ which consists of Eq. (19) and Eq. (24). The coefficients of the Eq. (19), (and the coefficients of the Eq. (24) respectively) are determined by the theorem 7. The response to the random action is defined completely.

We show how the stochastic invariants theory can be applied to solve different tasks.
