Arbitrary Coefficient Assignment by Static Output Feedback for Linear Differential Equations with Non-Commensurate Lumped and Distributed Delays

We consider a linear control system defined by a scalar stationary linear differential equation in the real or complex space with multiple non-commensurate lumped and distributed delays in the state. In the system, the input is a linear combination of multiple variables and its derivatives, and the output is a multidimensional vector of linear combinations of the state and its derivatives. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function by linear static output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for the solvability of the arbitrary coefficient assignment problem by the static output feedback controller. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We provide an example illustrating our results.

AdS3/AdS2 degression of massless particles

We study a 3d/2d dimensional degression which is a Kaluza-Klein type mechanism in AdS3 space foliated into AdS2 hypersurfaces. It is shown that an AdS3 massless particle of spin s = 1, 2, …, ∞ degresses into a couple of AdS2 particles of equal energies E = s. Note that the Kaluza-Klein spectra in higher dimensions are always infinite. To formulate the AdS3/AdS2 degression we consider branching rules for AdS3 isometry algebra o(2,2) representations decomposed with respect to AdS2 isometry algebra o(1,2). We find that a given o(2,2) higher-spin representation lying on the unitary bound (i.e. massless) decomposes into two equal o(1,2) modules. In the field-theoretical terms, this phenomenon is demonstrated for spin-2 and spin-3 free massless fields. The truncation to a finite spectrum can be seen by using particular mode expansions, (partial) diagonalizations, and identities specific to two dimensions.

Studying novel 1D potential via the AIM

In this work, we would like to apply the asymptotic iteration method (AIM) to a newly proposed Morse-like deformed potential introduced recently by Assi, Alhaidari and Bahlouli.[Formula: see text] This interesting potential can support bound states and/or resonances. However, in this work, we are only interested in bound states. We considered several choices of the potential parameters and obtained the associated spectrum. Finally, we study the small deformation limit at which this finite spectrum system will transition to infinite spectrum size.

Observation of Anderson localization beyond the spectrum of the disorder

Anderson localization is a fundamental wave phenomenon predicting that transport in a 1D uncorrelated disordered system comes to a complete halt, experiencing no transport whatsoever. However, in reality, a disordered physical system is always correlated, because it must have a finite spectrum. Common wisdom in the field states that localization is dominant only for wavepackets whose spectral extent resides within the region of the wavenumber span of the disorder. Here, we experimentally observe that Anderson localization can occur and even be dominant for wavepackets residing entirely outside the spectral extent of the disorder. We study the evolution of waves in synthetic photonic lattices containing bandwidth-limited (correlated) disorder, and observe Anderson localization for wavepackets of high wavenumbers centered around twice the mean wavenumber of the disorder spectrum. Likewise, we predict and observe Anderson localization at low wavenumbers, also outside the spectral extent of the disorder, and find that localization there can be as strong as for first-order transitions. This feature is universal, common to all Hermitian wave systems, implying that low-wavenumber wavepackets localize with a short localization length even when the disorder is strictly at high wavenumbers. This understanding suggests that disordered media should be opaque for long-wavelengths even when the disorder is strictly at much shorter length scales. Our results shed light on fundamental aspects of physical disordered systems and offer avenues for employing spectrally-shaped disorder for controlling transport in systems containing disorder.

Symmetry and unification from soft theorems and unitarity

We argue that symmetry and unification can emerge as byproducts of certain physical constraints on dynamical scattering. To accomplish this we parameterize a general Lorentz invariant, four-dimensional theory of massless and massive scalar fields coupled via arbitrary local interactions. Assuming perturbative unitarity and an Adler zero condition, we prove that any finite spectrum of massless and massive modes will necessarily unify at high energies into multiplets of a linearized symmetry. Certain generators of the symmetry algebra can be derived explicitly in terms of the spectrum and three-particle interactions. Furthermore, our assumptions imply that the coset space is symmetric.

The effects of tire dynamics on the performance of finite spectrum assignment of vehicle motion control

The lateral position control of the vehicle is analyzed in the presence of time delay. To compensate the negative effects of dead time, the predictor control approach called finite spectrum assignment is applied. This controller includes a linear model of the plant and uses the solution of this model over the delay interval to predict the current system states. The focus of the article is whether to include tire dynamics in the predictive model of the controller. Although the more detailed model should improve control performance, the additional parameters (e.g., tire stiffnesses and yaw moment of inertia) are difficult to determine accurately. The effects of parameter mismatches are analyzed in detail, and recommendations are given to ensure safe control of the vehicle. It is shown that the inclusion of tire dynamics in the predictive model vastly improves control performance even in the presence of large parameter errors, but in certain cases, the inaccuracies may lead to instability.

Finite spectrum assignment in linear systems with several lumped and distributed delays by means of static output feedback

We consider a control system defined by a linear time-invariant system of differential equations with lumped and distributed delays in the state variable. We construct a controller for the system as linear static output feedback with lumped and distributed delays in the same nodes. We study a finite spectrum assignment problem for the closed-loop system. One needs to construct gain coefficients such that the characteristic function of the closed-loop system becomes a polynomial with arbitrary preassigned coefficients. We obtain conditions on coefficients of the system under which the criterion was found for solvability of the finite spectrum assignment problem. Corollaries on stabilization by linear static output feedback with several delays are obtained for the closed-loop system.