Unstable Eigenvalues Research Articles

Asymptotic Representation for the Eigenvalues of a Non-selfadjoint Operator Governing the Dynamics of an Energy Harvesting Model

We consider a well known model of a piezoelectric energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler---Bernoulli model for the transverse vibrations of the beam and the second one represents the Kirchhoff's law for the electric circuit. Both equations are coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. Our main result is an explicit asymptotic formula for the eigenvalues of this generator, i.e., we perform the modal analysis for electrically loaded (not short-circuit) system. We show that the spectrum splits into an infinite sequence of stable eigenvalues that approaches a vertical line in the left half plane and possibly of a finite number of unstable eigenvalues. This paper is the first in a series of three works. In the second one we will prove that the generalized eigenvectors of the dynamics generator form a Riesz basis (and, moreover, a Bari basis) in the energy space. In the third paper we will apply the results of the first two to control problems for this model.

Spectral instability of general symmetric shear flows in a two-dimensional channel

In this paper, we prove the spectral instability of general symmetric shear flows of the incompressible Navier–Stokes equations at a high Reynolds number in a two-dimensional channel. This includes shear flows that are spectrally stable to the corresponding Euler equations, and thus for the first time, provides a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924) [5], C.C. Lin (1944) [9] and Tollmien (1947) [17], among others. Precisely, we construct exact unstable eigenvalues and eigenfunctions of the linearized Navier–Stokes equations around symmetric shear flows, showing that the solution could grow slowly at the rate of et/αR, where R is the sufficiently large Reynolds number and α is the small spatial frequency that remains between lower and upper marginal stability curves: αlow(R)≈R−1/7 and αup(R)≈R−1/11. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

Stability Optimization of a Disc Brake System with Hybrid Uncertainties for Squeal Reduction

A hybrid uncertain model is introduced to deal with the uncertainties existing in a disc brake system in this paper. By the hybrid uncertain model, the uncertain parameters of the brake with enough sampling data are treated as probabilistic variables, while the uncertain parameters with limited data are treated as interval probabilistic variables whose distribution parameters are expressed as interval variables. Based on the hybrid uncertain model, the reliability-based design optimization (RBDO) of a disc brake with hybrid uncertainties is proposed to explore the optimal design for squeal reduction. In the optimization, the surrogate model of the real part of domain unstable eigenvalue of the brake system is established, and the upper bound of its expectation is adopted as the optimization objective. The lower bounds of the functions related to system stability, the mass, and the stiffness of design component are adopted as the optimization constraints. The combinational algorithm of Genetic Algorithm and Monte-Carlo method is employed to perform the optimization. The results of a numerical example demonstrate the effectiveness of the proposed optimization on improving system stability and reducing squeal propensity of a disc brake under hybrid uncertainties.

On the tightness of linear policies for stabilization of linear systems over Gaussian networks

In this paper, we consider stabilization of multi-dimensional linear systems driven by Gaussian noise controlled over parallel Gaussian channels. For such systems, it has been recognized that for stabilization in the sense of asymptotic stationarity or stability in probability, Shannon capacity of a channel is an appropriate measure on characterizing whether a system can be made stable when controlled over the channel. However, this is in general not the case for quadratic stabilization. On a related problem of joint-source channel coding, in the information theory literature, the source-channel matching principle has been shown to lead to optimality of uncoded or analog transmission and when such matching conditions occur, it has been shown that capacity is also a relevant figure of merit for quadratic stabilization. A special case of this result is applicable to a scalar LQG system controlled over a scalar Gaussian channel. In this paper, we show that even in the absence of source-channel matching, to achieve quadratic stability, it may suffice that information capacity (in Shannon’s sense) is greater than the sum of the logarithm of unstable eigenvalue magnitudes. In particular, we show that periodic linear time varying coding policies are optimal in the sense of obtaining a finite second moment for the state of the system with minimum transmit power requirements for a large class of vector Gaussian channels. Our findings also extend the literature which has considered noise-free systems.

Instability Analysis of Uncertain Systems via Determinants and LMIs

This technical note investigates two instability measures in continuous-time (CT) and discrete-time (DT) uncertain systems, the first given by the spectral abscissa (CT case) or radius (DT case), and the second given by the sum (CT case) or product (DT case) of the unstable eigenvalues. It is supposed that the system depends polynomially on an uncertain vector constrained into a semi-algebraic set. The problem is to determine the largest instability measures over the admissible uncertainties. It is shown that a sufficient condition for establishing an upper bound of the sought measures can be obtained in terms of linear matrix inequality (LMI) feasibility tests by exploiting the determinants of some specific matrices, and that this condition is also necessary under some mild conditions on the semi-algebraic set by using multipliers with degree sufficiently large. Moreover, a condition is provided for establishing the tightness of the found best upper bounds. Lastly, it is shown that in the special case where the semi-algebraic set is an interval, the degree of the multipliers is known a priori. Some numerical examples illustrate the proposed results.

Quantifying the unstable in linearized nonlinear systems

It has been shown that quantifying the unstable in linear systems is important for establishing the existence of stabilizing feedback controllers in the presence of communications constraints. In this context, the instability measure is defined as the sum of the real parts (continuous-time case) or the product of the magnitudes (discrete-time case) of the unstable eigenvalues. This paper addresses the problem of quantifying the unstable in linearized systems obtained from nonlinear systems for a family of constant inputs, i.e., quantifying the largest instability measure over all admissible equilibrium points and all admissible constant inputs. It is supposed that the dynamics of the nonlinear system is polynomial in both state and input, either continuous-time or discrete-time, and that the set of constant inputs is a semialgebraic set. Two cases are considered: first, when the equilibrium points are known polynomial functions of the input, and, second, when the equilibrium points are unknown (polynomial or non-polynomial) functions of the input. It is shown that upper bounds of the sought instability measure can be established through linear matrix inequalities (LMIs) by searching for polynomially-dependent Lyapunov function candidates. Moreover, it is shown that these upper bounds are nonconservative for a sufficiently large degree of the Lyapunov function candidates under some conditions. Lastly, necessary and sufficient conditions are provided for establishing whether the obtained upper bounds are nonconservative. Some numerical examples also show the advantages of the proposed method with respect to grid techniques.

Stabilization of networked control systems with both network-induced delay and packet dropout

This paper is concerned with the mean-square stabilization problem for discrete-time networked control systems (NCSs). It is assumed that control signal is sent to plant over a lossy communication channel, where network-induced delay and packet dropout occur simultaneously. A necessary and sufficient stabilizing condition is developed in terms of the unique positive-definite solutions to some coupled algebraic Riccati equations (CAREs). The contributions of this paper are twofold. First, an existence theorem of the maximum packet dropout rate is proposed. Second, for one-dimensional single-input system and the decoupled multi-input system, it is shown that the NCS is stabilizable iff the network-induced delay and the packet dropout rate satisfy some simple algebraic inequalities. If the network-induced delay is known a priori, the maximum packet dropout rate is given explicitly in terms of network-induced delay and unstable eigenvalues of the system matrix. If the packet dropout rate is known a priori, the maximum allowable delay bound is also given explicitly.

Numerical evidence for universality in the excited instability spectrum of magnetically charged Reissner–Nordström black holes

It is well known that the SU(2) Reissner–Nordström black-hole solutions of the Einstein–Yang–Mills theory are characterized by an infinite set of unstable (imaginary) eigenvalues \(\{\omega _n(T_{\text {BH}})\}_{n=0}^{n=\infty }\) (here \(T_{\text {BH}}\) is the black-hole temperature). In this paper we analyze the excited instability spectrum of these magnetically charged black holes. The numerical results suggest the existence of a universal behavior for these black-hole excited eigenvalues. In particular, we show that unstable eigenvalues in the regime \(\omega _n\ll T_{\text {BH}}\) are characterized, to a very good degree of accuracy, by the simple universal relation \(\omega _n(r_+-r_-)={\text {constant}}\), where \(r_{\pm }\) are the horizon radii of the black hole.

Strong exchange of stabilities in rapidly rotating parabolic Poiseuille flow

Abstract Poiseuille flow between two coaxial cylindrical pipes, with inner pipe translating at just the rate so that the velocity is parabolic, is rotated around the common axis of symmetry, with a common angular velocity. If the rotation rate is very large and it is assumed that the wave length of the critical disturbances is small, the governing equations take an approximate form that is analogous, remarkably, to Couette flow between coaxial cylinders with a certain difference. It is proved that if the rotation rate meets this criterion, the strong principle of exchange of stabilities (PES) holds; that is, all eigenvalues are real, and there is a first unstable eigenvalue. From a physical viewpoint, the resulting unstable motion initially takes the form of steady spiral vortices.

Symmetry breaking of solitons in two-dimensional complex potentials.

Symmetry breaking is reported for continuous families of solitons in the nonlinear Schrödinger equation with a two-dimensional complex potential. This symmetry breaking is forbidden in generic complex potentials. However, for a special class of partially parity-time-symmetric potentials, it is allowed. At the bifurcation point, two branches of asymmetric solitons bifurcate out from the base branch of symmetry-unbroken solitons. Stability of these solitons near the bifurcation point are also studied, and two novel properties for the bifurcated asymmetric solitons are revealed. One is that at the bifurcation point, zero and simple imaginary linear-stability eigenvalues of asymmetric solitons can move directly into the complex plane and create oscillatory instability. The other is that the two bifurcated asymmetric solitons, even though having identical powers and being related to each other by spatial mirror reflection, can possess different types of unstable eigenvalues and thus exhibit nonreciprocal nonlinear evolutions under random-noise perturbations.

Network-Decentralized Control Strategies for Stabilization

We consider the problem of stabilizing a class of systems formed by a set of decoupled subsystems (nodes) interconnected through a set of controllers (arcs). Controllers are network-decentralized, i.e., they use information exclusively from the nodes they interconnect. This condition requires a block-structured feedback matrix, having the same structure as the transpose of the overall input matrix of the system. If the subsystems do not have common unstable eigenvalues, we demonstrate that the problem is solvable. In the general case, we provide sufficient conditions for solvability. When subsystems are identical and each input agent controls a pair of subsystems with input matrices having opposite sign (flow networks), we prove that stabilization is possible if and only if the system is connected with the external environment. Our proofs are constructive and lead to structured linear matrix inequalities (LMIs).

Stability Analysis for Combustion Fronts Traveling in Hydraulically Resistant Porous Media

We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively unstable.

Variational necessary and sufficient stability conditions for inviscid shear flow.

A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle, where the velocity profile is assumed to be monotonic and analytic. It is shown that unstable eigenvalues of Rayleigh's equation (which is a non-self-adjoint eigenvalue problem) can be associated with positive eigenvalues of a certain self-adjoint operator. The stability is therefore determined by maximizing a quadratic form, which is theoretically and numerically more tractable than directly solving Rayleigh's equation. This variational stability criterion is based on the understanding of Kreĭn signature for continuous spectra and is applicable to other stability problems of infinite-dimensional Hamiltonian systems.

The instability spectrum of weakly-magnetized SU(2) Reissner–Nordström black holes

It is well known that the U(1) Reissner–Nordström black hole is stable within the framework of the Einstein–Maxwell theory. However, the SU(2) Reissner–Nordström black-hole solution of the coupled Einstein–Yang–Mills equations is known to be unstable. In fact, this magnetically charged black hole is characterized by an infinite set of unstable (growing in time) perturbation modes. In the present paper we study analytically the instability resonance spectrum of weakly-magnetized SU(2) Reissner–Nordström black holes. In particular, we obtain explicit analytical expressions for the infinite set {ωn}n=0n=∞ of imaginary eigenvalues that characterize the instability growth rates of the perturbation modes. We discuss the role played by these unstable eigenvalues as critical exponents in the gravitational collapse of the Yang–Mills field. Finally, it is shown that our analytical formulas for the characteristic black-hole instability spectrum agree with new numerical data that recently appeared in the literature.

Minimality of State Space Solutions of DSGE Models and Existence Conditions for Their VAR Representation

Abstract Standard solution methods of DSGE models do not necessarily deliver minimal state space forms. When the ABCD form is non-minimal, the conditions in the literature are not necessary for the existence of a VAR representation of the observables. In this paper we present necessary and sufficient conditions that are valid in general, and hence can be applied to minimal and non-minimal ABCD forms. If the state space form is minimal, our conditions coincide with those in the literature. These results also clarify that it is possible to have unstable eigenvalues together with a (possibly finite) VAR representation of the observables.

Stabilizability Across a Gaussian Product Channel: Necessary and Sufficient Conditions

We present necessary and sufficient conditions for stabilizing a discrete-time LTI plant in the mean squared sense when a sensor transmits the plant state information to a remotely placed controller across a Gaussian product channel. The Gaussian product channel models a continuous-time waveform Gaussian channel, where the encoder transmits information to the receiver across multiple noisy paths. The conditions presented are in terms of the power at the transmitter, noise variances and the unstable eigenvalues of the plant. It is known that linear coding schemes may lead to overly restrictive stabilizability conditions in such scenarios. We present a nonlinear coding scheme and present the resulting sufficient stabilizability conditions. We then prove the necessity of the conditions using information theoretic tools.

Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: Application to open-loop control

A fully discrete formalism is introduced to perform stability analysis of a turbulent compressible flow whose dynamics is modeled with the Reynolds-Averaged Navier–Stokes (RANS) equations. The discrete equations are linearized using finite differences and the Jacobian is computed using repeated evaluation of the residuals. Stability of the flow is assessed solving an eigenvalue problem. The sensitivity gradients which indicate regions of the flow where a passive control device could stabilize the unstable eigenvalues are defined within this fully discrete framework. The second order finite differences are applied to the discrete residual to compute the gradients. In particular, the sensitivity gradients are shown to be linked to the Hessian of the RANS equations. The introduced formalism and linearization method are generic: the code used to evaluate the residual of the RANS equations can be used in a black box manner, and the complex linearization of the Hessian is avoided. The method is tested on a two dimensional deep cavity case, the flow is turbulent with a Reynolds number equal to 860 000 and compressible with a Mach number of 0.8. Several turbulence models and numerical schemes are used to validate the method. Physical features of the flow are recovered, such as the fundamental frequency of the natural flow as well as acoustic mechanisms, suggesting the validity of the method. The sensitivity gradients are then computed and validated, the error in predicting the eigenvalue variation being found less than 3%. Control maps using a small steady control device are finally obtained, indicating that the control area should be chosen in the vicinity of the leading edge of the cavity.

Stability of an inflated hyperelastic membrane tube with localized wall thinning

It is now well-known that when an infinitely long hyperelastic membrane tube free from any imperfections is inflated, a transcritical-type bifurcation may take place that corresponds to the sudden formation of a localized bulge. When the membrane tube is subjected to localized wall-thinning, the bifurcation curve would “unfold” into the turning-point type with the lower branch corresponding to uniform inflation in the absence of imperfections, and the upper branch to bifurcated states with larger amplitude. In this paper stability of bulged configurations corresponding to both branches is investigated with the use of the spectral method. It is shown that under pressure control and with respect to axi-symmetric perturbations, configurations corresponding to the lower branch are stable but those corresponding to the upper branch are unstable. Stability or instability is established by demonstrating the non-existence or existence of an unstable eigenvalue (an eigenvalue with a positive real part). This is achieved by constructing the Evans function that depends only on the spectral parameter. This function is analytic in the right half of the complex plane where its zeroes correspond to the unstable eigenvalues of the generalized spectral problem governing spectral instability. We show that due to the fact that the skew-symmetric operator J involved in the Hamiltonian formulation of the basic equations is onto, the zeroes of the Evans function can only be located on the real axis of the complex plane. We also comment on the connection between spectral (linear) stability and nonlinear (Lyapunov) stability.

Equivalent conditions on periodic feedback stabilization for linear periodic evolution equations

This paper studies the periodic feedback stabilization for a class of linear T-periodic evolution equations. Several equivalent conditions on the linear periodic feedback stabilization are obtained. These conditions are related to the following subjects: the attainable subspace of the controlled evolution equation under consideration; the unstable subspace (of the evolution equation with the null control) provided by the Kato projection; the Poincaré map associated with the evolution equation with the null control; and two unique continuation properties for the dual equations on different time horizons [0,T] and [0,n0T] (where n0 is the sum of algebraic multiplicities of distinct unstable eigenvalues of the Poincaré map). It is also proved that a T-periodic controlled evolution equation is linear T-periodic feedback stabilizable if and only if it is linear T-periodic feedback stabilizable with respect to a finite-dimensional subspace. Some applications to heat equations with time-periodic potentials are presented.

Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment

A new approach for stabilizing unstable reduced order models (ROMs) for linear time-invariant (LTI) systems through an a posteriori post-processing step applied to the algebraic ROM system is developed. The key idea is to modify the unstable eigenvalues of the ROM system by moving these eigenvalues into the stable half of the complex plane. It is demonstrated that this modification to the ROM system eigenvalues can be accomplished using full state feedback (a.k.a. pole placement) algorithms from control theory. This approach ensures that the modified ROM is stable provided the system’s unstable poles are controllable and observable; however, the accuracy of the stabilized ROM is not guaranteed. To remedy this difficulty and guarantee an accurate stabilized ROM, a constrained nonlinear least-squares optimization problem for the stabilized ROM eigenvalues in which the error in the ROM output is minimized is formulated. This optimization problem is small and therefore computationally inexpensive to solve. Performance of the proposed algorithms is evaluated on two test cases for which ROMs constructed via the proper orthogonal decomposition (POD)/Galerkin method suffer from instabilities.