Mobility limits of the Lyapunov exponents of linear systems under small average perturbations

Mobility limits of the Lyapunov and central exponents of linear systems of differential equations, under arbitrarily small average linear perturbations are found with the aid of V. M. Millionshchikov's method of rotations. One obtains stability criteria for these exponents with respect to the mentioned perturbations, as well as criteria for the stabilizability or destabilizability of the zero solution.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 32–73, 1986.     

Nondiagonable systems under small perturbations

By a nondiagonable system, we mean a system whose state matrix is nondiagonable, i.e. having nonlinear elementary divisors. In this paper, closed form formulae are given for the shifts occuring in the eigenvalues and eigenvectors of these types of systems due to small variations in the system elements.

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Zusammenfassung Ein System heißt nicht-diagonalisierbar, wenn seine Zustandsmatrix nichtlineare Elementarteiler hat In dieser Arbeit werden geschlossene Formeln für die Änderungen der Eigenwerte und der Eigenvektoren eines nicht-diagonalisierbaren Systems angegeben, welche durch kleine Änderungen in den Systemelementen verursacht werden.

     

Stability and robustness analysis of a linear time-periodic system subjected to random perturbations

In this work, new methods of guaranteeing the stability of linear time periodic dynamical systems with stochastic perturbations are presented. In the approaches presented here, the Lyapunov-Floquet (L-F) transformation is applied first so that the linear time-periodic part of the equations becomes time-invariant. For the linear time periodic system with stochastic perturbations, a stability theorem and related corollary have been suggested using the results previously obtained by Infante. This technique is not only applicable to systems with stochastic parameters but also to systems with deterministic variation in parameters. Some illustrative examples are presented to show the practical applications. These methods can be used to investigate the degree of robustness and design controllers for systems with time periodic coefficients subjected to random perturbations.     

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ε ^?1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).     

On perturbations of controllable implicit linear systems

Various notions of controllability for general (i.e. possiblynon-regular) implicit linear systems are considered. Some genericityproperties of the corresponding controllable systems are established.     

Reconstruction of boundary controls in parabolic systems

In the paper, an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamical systems described by boundary value problems for partial differential equations of parabolic type. The source information for solving the inverse problem is the results of approximate measurements of the states of the observed system’s motion. The problem is solved in the static case; i.e., to solve it, we use all the measurement data accumulated during some specified observation interval. The problem under consideration is ill-posed. To solve it, we propose the Tikhonov method with a stabilizer containing the sum of the mean-square norm and total time variation of the control. The use of such nondifferentiable stabilizer allows us to obtain more precise results than the approximation of the desired control in the Lebesgue spaces. In particular, this method provides the pointwise and piecewise uniform convergences of regularized approximations and makes possible the numerical reconstruction of the subtle structure of the desired control. In the paper, the subgradient projection method for obtaining a minimizing sequence for the Tikhonov functional is described and substantiated. Also, we demonstrate the two-stage finitedimensional approximation of the problem and present the results of numerical simulation.     

Uniformly exponentially stable approximations for a class of damped systems

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters.     

Exact boundaries of upper mobility of the Lyapunov exponents of linear differential systems under exponentially decaying perturbations of the coefficient matrices

For a linear differential system, we obtain formulas for the computation of the exact boundaries of upper mobility for Lyapunov exponents under exponentially decaying perturbations of its coefficient matrix on the basis of the Cauchy matrix.     

Robust controller for systems with exponentially stable strongly continuous semigroups

The theory of robust controllers is extended to the case where we have boundary and/or distributed control and the system operator is an infinitesimal generator of a strongly continuous exponentially stable semigroup. An example on hyperbolic systems is presented.     

Optimal control for distributed linear systems subjected to null-controllability

In this article, we apply the notion of hierarchic control on a distributed system in which the state is governed by a parabolic equation. This notion assumes that we have two controls where one will be the Leader and the other, the Follower. The first control is of controllability type subjected to a constraint, while the second expresses that the state does not move too far from a given state. The results are achieved by means of an observability inequality of the Carleman type, which is ‘adapted’ to the constraint.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Sharp bounds on mobility of the Lyapunov exponents of linear Hamiltonian systems under small average perturbations

Sharp bounds on mobility of the Lyapunov exponents of linear Hamiltonian systems are found for arbitrarily small average linear perturbations of the coefficients, with the help of the Millionshchikov turning method. The stabilizability and destabilizability of any solution by the indicated perturbations in the class of linear Hamiltonian systems are proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 125–139, 1989.     

On uniformly stable linear quasiperiodic systems

In a finite-dimensional complex space, we consider a system of linear differential equations with quasiperiodic skew-Hermitian matrix. The space of solutions of this system is a sum of one-dimensional invariant subspaces. Over a torus defined by a quasiperiodic matrix of the system, we investigate the corresponding one-dimensional invariant bundles (nontrivial in the general case). We find conditions under which these bundles are trivial and the system can be reduced to diagonal form by means of the Lyapunov quasiperiodic transformation with a frequency module coinciding with the frequency module of the matrix of the system.     

Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l _??(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel?CLegendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system??s data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et?al. (SIAM J. Optim. 20, 1504?C1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system??s coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.     

Identification of linear nonstationary systems from their responses to generalized controls

We suggest a method for reconstructing the coefficients of a linear nonstationary system from its responses to generalized controls (delta functions and their generalized derivatives).     

The efficiency of perturbations in the class of linear Hamiltonian systems

It is proved that the set of all limiting values of solutions’ arbitrary indicators under uniformly small perturbations of coefficients of a linear Hamiltonian system is the same as the similar set obtained by uniformly small Hamiltonian perturbations.