Stable invariant manifolds with application to control problems

<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>

Calculation of Invariant Manifolds of Piecewise-Smooth Maps

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.

Stable manifolds for perturbations of exponential dichotomies in mean

We establish the existence of stable invariant manifolds for any sufficiently small perturbation of a cocycle with an exponential dichotomy in mean. The latter notion corresponds to replace the exponential behavior in the classical notion of an exponential dichotomy by an exponential behavior in average with respect to an invariant measure. We consider both perturbations of a cocycle over a map and over a flow that can be defined on an arbitrary Banach space. Moreover, we obtain an upper bound for the speed of the nonlinear dynamics along the stable manifold as well as a lower bound when the exponential dichotomy in mean is strong (this means that we have lower and upper bounds along the stable and unstable directions of the dichotomy).

Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under(μ,ν)-Dichotomies

We obtain the existence of stable invariant manifolds for the nonlinear equationx′=L(t)xt+f(t,xt,λ)provided that the linear delay equationx′=L(t)xtadmits a nonuniform(μ,ν)-dichotomy andfis a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameterλ. Namely, the stable invariant manifolds are Lipschitz in the parameterλ. In addition, we also show that nonuniform(μ,ν)-contraction persists under sufficiently small nonlinear perturbations.