The application of the gap metric to robust stabilization of feedback systems is considered. In particular, a solution to the problem of robustness optimization in the gap metric is presented. The problem of robust stabilization under simultaneous plant-controller perturbations is addressed, and the least amount of combined plant-controller uncertainty, measured by the gap metric, that can cause instability of a nominally stable feedback system is determined. Included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the T-gap metric. A key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations. This settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well-defined metric. >

Optimal robustness in the gap metric

The author presents results regarding certain average properties of incompressible fluids derived from the equations of motion. The author estimates the average dissipation rate, the average dimension of level sets. The role played by the field of direction of vorticity in the three-dimensional Euler and Navier-Stokes equations is discussed and a class of two-dimensional equations that are useful models of incompressible dynamics is described. The author presents results concerning scaling exponents in turbulence.

Geometric statistics in turbulence

Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the RayleighTaylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

https://web.math.princeton.edu/~const/eule.pdf

On the Euler equations of incompressible fluids

It is shown that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations. The proof of this result involves showing that a ball of uncertainty in the gap metric, of a given radius, is equal to a ball of uncertainty, of the same radius, defined by perturbations of a normalized right coprime fraction, provided the radius is sufficiently small for a controller to be found to stabilize the ball. The relationship between balls of uncertainty defined by perturbations of left fractions with those defined by perturbations of right fractions is investigated. It is shown that these are, in general, different, although it turns out that a controller stabilizes all plants in the left ball if and only if it stabilizes all plants in the right ball. The problem of combined plant and controller uncertainty is also addressed. The plant and controller are shown to play reciprocal roles with respect to uncertainty in the gap. The minimal amount of combined plant and controller uncertainty that may cause instability is determined. A detailed summary of the main properties of the gap metric is given, and a dual metric called the T-gap metric is introduced. The extension of the results to infinite dimensional systems is discussed. >

Bounds on the bulk rate of energy dissipation in body-force-driven steady-state turbulence are derived directly from the incompressible Navier–Stokes equations We consider flows in three spatial dimensions in the absence of boundaries and derive rigorous a priori estimates for the time-averaged energy dissipation rate per unit mass, e, without making any further assumptions on the flows or turbulent fluctuations We provee [les ] c1v U2/l2 + c2 U3/l,where v is the kinematic viscosity, U is the root-mean-square (space and time averaged) velocity, and l is the longest length scale in the applied forcing function The prefactors c1 and c2 depend only on the functional shape of the body force and not on its magnitude or any other length scales in the force, the domain or the flow We also derive a new lower bound on e in terms of the magnitude of the driving force F For large Grashof number Gr = Fl3/v2, we findc3 vFl/λ2 [les ] ewhere λ = √vU2/e is the Taylor microscale in the flow and the coefficient c3 depends only on the shape of the body force This estimate is seen to be sharp for particular forcing functions producing steady flows with λ/l ∼ O(1) as Gr → 1 We interpret both the upper and lower bounds on e in terms of the conventional scaling theory of turbulence – where they are seen to be saturated – and discuss them in the context of experiments and direct numerical simulations

/pdf/energy-dissipation-in-body-forced-turbulence-2g7subt4rb.pdf

Energy dissipation in body-forced turbulence

On the statistical study of the Navier-Stokes equations

/pdf/on-contractions-similar-to-isometries-and-toeplitz-operators-5fs9mhv8zp.pdf

On contractions similar to isometries and Toeplitz operators

On the Lax-Phillips nonconservative scattering theory

Local characterization of operators of class C0

It is proved that if T is a completely nonunitary contraction on Hilbert space such that T*` does not converge strongly to 0 as n 0 oo, there is an integer N > 0 so that none of the powers T*' with m N has a cyclic vector. Both conditions on T are essential, and the integer N is not universal, i.e., it depends on T. In this note we are concerned with contractions T of Hilbert space 'X. In previous papers [3] we considered conditions which assure the existence of cyclic vectors for T (or T*). Here we shall deal with a converse question, namely, to find contractions without cyclic vectors. 1. First let us recall [3 or 4, Chapter II] that if U and U+ are the minimal unitary and isometric dilations of T, acting on the spaces X and '}Q1, respectively (SC C SC+ C XC), and if 1 is the subspace of T+ on which U+ is unitary, i.e., the subspace 00 ( 1) 6g = fl'j + U 0 n-O then 6R reduces both U+ and U to the unitary operator R = U+ V '. Moreover, setting X = PGR I C and tX. = PR DC (where Pq denotes the orthogonal projection of D onto 'R ), we have (2) XT* = R*X and R* (R*X )= (XT*DC)C (X'X)= 'A; and hence, (3) OX C RAX. Because of the minimality of U+ , and since 4. reduces U+ to R, we also have

https://www.ams.org/proc/1983-087-04/S0002-9939-1983-0687638-1/S0002-9939-1983-0687638-1.pdf

Ciprian Foiaş

Papers

On the statistical study of the Navier-Stokes equations

On contractions similar to isometries and Toeplitz operators

On the Lax-Phillips nonconservative scattering theory

Local characterization of operators of class C0

Contractions without cyclic vectors