scholarly journals Stable invariant manifolds with application to control problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Alexey Gorshkov
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Burgers Equation ◽  
Linear Equations ◽  
Control Problems ◽  
Stable Invariant Manifolds ◽  
Global Invariant ◽  
Stable Invariant ◽  
Discrete And Continuous Spectrum ◽  
Non Linear Equations

<p style='text-indent:20px;'>In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation.</p>

scholarly journals Calculation of Invariant Manifolds of Piecewise-Smooth Maps

2020 ◽  
Vol 24 (3) ◽  
pp. 166-182
Author(s):  
Z. T. Zhusubaliyev ◽  
V. G. Rubanov ◽  
Yu. A. Gol’tsov
Keyword(s):  
Invariant Manifolds ◽  
Inverse Function ◽  
Piecewise Smooth ◽  
Smooth Maps ◽  
First Order ◽  
Piecewise Smooth Maps ◽  
Stable Invariant Manifolds ◽  
Stable Invariant ◽  
Original Approach ◽  
Stable Subspace

Purpose of reseach is of the work is to develop an algorithm for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps. Method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. Results. A method for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps is developed. The main result is formulated as a statement. The method is based on an original approach to finding the inverse function, the idea of which is to reduce the problem to a nonlinear first-order equation. Conclusion. A numerical method is described for calculating stable invariant manifolds of piecewise smooth maps that simulate impulse automatic control systems. The method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. The method is based on an original approach to finding the inverse function, which consists in reducing the problem to solving a nonlinear first-order equation. This approach eliminates the need to solve systems of nonlinear equations to determine the inverse function and overcome the accompanying computational problems. Examples of studying the global dynamics of piecewise-smooth mappings with multistable behavior are given.

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