Stabilization of solutions of the stochastic Dzekzer equation

Evgeny D. Akhmadeev, Olga G. Kitaeva

Abstract


The article considers the stochastic Dzekzer equation, which describes the evolution of the free surface of a filtered liquid. To study the stability and instability of solutions and the stabilization of unstable solutions, this equation in suitable functional stochastic spaces is considered as a linear stochastic equation of the Sobolev type. The solution to the stochastic equation is a stochastic process that is not differentiable by Newton - Leibniz at any point. Therefore, we use the derivative of the stochastic process in the sense of Nelson - Gliklikh.  The question of stability and instability of solutions to the stochastic Dzekzer equation is solved in terms of stable and unstable invariant spaces. To solve the stabilization problem, we consider the stochastic equation of the Sobolev type as a system of three equations: one singular and two regular, defined on stable and unstable invariant spaces. With the help of a feedback loop, the problem of stabilizing unstable solutions is solved. A numerical experiment has been carried out. Graphs of the solution before stabilization and after stabilization are given.

Keywords


Sobolev type equations; stochastic Dzekzer equation; invariant spaces; stabilization problem

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References


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