Preservation of stability of dynamical systems under homomorphisms

For abstract dynamical systems in the form of systems of motions, we obtain criteria for the preservation of stability properties via an extended notion of a homomorphism of dynamical systems. We obtain corollaries for ordinary differential equations including those with switchings of vector fields. We present examples illustrating the use of these criteria.     

Relaxed Euler systems and convergence to Navier-Stokes equations

We consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation both in compressible and incompressible cases. This requires to decompose the second-order derivative terms of the velocity into first-order ones. Usual decompositions lead to approximate systems with tensor variables. We construct approximate systems with vector variables by using Hurwitz-Radon matrices. These systems are written in the form of balance laws and admit strictly convex entropies, so that they are symmetrizable hyperbolic. For smooth solutions, we prove the convergence of the approximate systems to the Navier-Stokes equations in uniform time intervals. Global-in-time convergence is also shown for the initial data near constant equilibrium states of the systems. These convergence results are established not only for the approximate systems with vector variables but also for those with tensor variables.     

System of nonsmooth variational inequalities with applications

In this paper, we consider a system of vector variational inequalities and a system of nonsmooth variational inequalities defined by means of Clarke directional derivative. We also consider the Nash equilibrium problem with vector pay-offs and its scalarized form. We present some relations among these systems and problems. The existence results for a solution of system of nonsmooth variational inequalities are given. As a consequence, we derive an existence result for a solution of Nash equilibrium problem with vector pay-offs.     

Vector hyperbolic equations with higher symmetries

We list eleven vector hyperbolic equations that have third-order symmetries with respect to both characteristics. This list exhausts the equations with at least one symmetry of a divergence form. We integrate four equations in the list explicitly, bring one to a linear form, and bring four more to nonlinear ordinary nonautonomous systems. We find the Bäcklund transformations for six equations.     

Generalization of the Jurdjevic-Quinn theorem and stabilization of bilinear control systems with periodic coefficients: I

The Jurdjevic-Quinn theorem on the global asymptotic stabilization of the origin is generalized to nonlinear time-varying affine control systems with periodic coefficients. The proof is based on the Krasovskii theorem on the global asymptotic stability for periodic systems and the introduced notion of “commutator” for two vector fields one of which is time-varying. The obtained sufficient conditions for stabilization are applied to bilinear control systems with periodic coefficients. We construct a control periodic in t in the form of a quadratic form in x that asymptotically stabilizes the zero solution of a bilinear periodic system with a time-invariant drift.     

On stability of interconnected finite difference systems

We give a method of construction of Lyapunov functions in the form of a linear form with respect to moduli of variables, for which there exist Krasovskii constants in the case of asymptotic stability, for linear systems with constant coefficients and some types of nonlinear systems of finite-difference equations. An application of the above functions as components of a vector Lyapunov function allowed us to obtain conditions on asymptotic stability for interrelated finite-difference systems.Translated from Dinamicheskie Sistemy, No. 8, pp. 68–71, 1989.     

On the construction of solutions of terminal problems for multidimensional affine systems in quasicanonical form

We consider the construction of solutions of terminal problems for multidimensional affine systems. We show that the terminal problem for a regular system in quasicanonical form can be reduced to a boundary value problem for a system of ordinary differential equations of lower order with right-hand side depending on a vector parameter. We prove a sufficient condition for the existence of a solution of the above-mentioned boundary value problem. A method for constructing a numerical solution is developed.     

Embedding and splitting ordinary differential equations in normal form

We introduce an embedding of real or complex n-dimensional space Kⁿ as an algebraic variety V which is determined by the action of a linear one-parameter group. Every analytic vector field on Kⁿ corresponds to some embedded vector field on V. For a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non-autonomous linear systems. Examples and applications are mostly concerned with Poincaré-Dulac normal forms. Embeddings provide a natural setting for perturbations of symmetric systems, in particular of systems in normal form up to some degree.     

Minimum-order observers for discrete-time systems

We consider the synthesis of a minimum-order state or functional observer for a linear dynamical system. The synthesis problem is solved for completely certain systems of general form and for some classes of uncertain systems. Various approaches are described, which ultimately lead to the same task: finding a minimum-dimension Hurwitz solution for a system of linear equations with a Hankel matrix. For scalar and vector linear systems, prior upper and lower bounds on the observer dimension are derived, which makes it possible to switch to an iterative procedure of finding an optimal solution. The discussion is set out for discrete-time dynamical systems.     

The Dependence of Optimal Returns from Multi-class Queueing Systems on Their Customer Base

We identify structured collections of multi-class queueing systems whose optimal return (a minimised cost) is a supermodular function of the set of customer classes allowed external access to the system. Our results extend considerably the range of systems for which such a claim can be made. The returns from such systems also exhibit a form of directional convexity when viewed as functions of a vector of arrival rates. Applications to load balancing problems are indicated.     

Nonlinear Diffusion Processes

We study elliptic systems of strongly nonlinear first-order differential equations in complex form on the plane. For such systems we develop the theory of Hilbert boundary value problems which is very much similar to the well-known theory for a holomorphic vector. Systems of nonlinear elliptic equations describe problems of interaction of several nonlinear stationary processes in the diffusive and convective mass and heat transport by hydrodynamic fluid flows.     

Hamilton-Jacobi inequalities in control problems for impulsive dynamical systems

We propose definitions of strong and weak monotonicity of Lyapunov-type functions for nonlinear impulsive dynamical systems that admit vector measures as controls and have trajectories of bounded variation. We formulate infinitesimal conditions for the strong and weak monotonicity in the form of systems of proximal Hamilton-Jacobi inequalities. As an application of strongly and weakly monotone Lyapunov-type functions, we consider estimates for integral funnels of impulsive systems as well as necessary and sufficient conditions of global optimality corresponding to the approach of the canonical Hamilton-Jacobi theory.     

Robust inversion algorithms for vector linear systems

We consider the inversion problem for linear systems, which involves estimation of the unknown input vector. The inversion problem is considered for a system with a vector output and a vector input assuming that the observed output is of higher dimension than the unknown input. The problem is solved by using a controlled model in which the control stabilizes the deviations of the model output from the system output. The stabilizing model control or its averaged form may be used as the estimate of the unknown system input. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 17–22, 2004.     

A point-wise criterion for quasi-periodic motions in the KAM theory

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rüssmann’s nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.     

Reducing triangular systems of ODEs with rational coefficients,with applications to coupled Regge-Wheeler equations

We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operators with rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, which have naturally appeared in previous work on vector and tensor perturbations on the Schwarzschild black hole spacetime. Our systematic approach reproduces and complements identities that have been previously found by trial and error methods.     

Hermite‐Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports

Hermite‐Padé approximants of type II are vectors of rational functions with a common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szeg?‐type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann‐Hilbert problem for multiple orthogonal polynomials (the common denominator).© 2016 Wiley Periodicals, Inc.     

Control of Hopf bifurcations of nonlinear infinite-dimensional systems: application to axial flow engine compressor

We study the local feedback stabilization of Hopf bifurcations for nonlinear systems of infinite dimensions in the case where the linearized vector field has a pair of simple nonzero imaginary eigenvalues and all its other eigenvalues lie strictly in the left half-plane. Discussing the normal form of nonlinear systems obtained by making use of the integral averaging method, we obtain sufficient and necessary condition for controlling the stability of the systems even if the critical modes are uncontrollable. As an application, we apply the obtained results to the control of axial flow engine compressor.     

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).     

Two-dimensional Fuchsian systems and the Chebyshev property

Let (x(t),y(t))^? be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P(t)x(t)+Q(t)y(t), where P,Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.     

Normalization in Lie algebras via mould calculus and applications

We establish Écalle’s mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré–Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.