Semidefinite relaxation of a class of quadratic integral inequalities

Fantuzzi, Giovanni; Wynn, Andrew

doi:10.1109/cdc.2016.7799221

Cited by 5 publications

(15 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…with V d (u) defined as in (17), S(u) defined as in (30), and R (u) defined as in (24). Therefore, we may use (38) and (39) to conclude…”

Section: Resultsmentioning

confidence: 99%

“…where V and V d are defined in (16) and (17), respectively. Then, there exists a positive scalar κ such that the solutions of (8) satisfy…”

Section: Problem Statementmentioning

confidence: 99%

“…where h1 is defined in (50). Using these functions and the definitions in (46), we may apply (17) to compute…”

Section: A Boundary Controlled Parabolic Pde-backstepping Feedback Lawmentioning

confidence: 99%

“…Recently, a number of direct methods for stability analysis were proposed with their respective numerical formulations [39], [33]. While avoiding the truncation of the PDE dynamics, these methods impose a set of basis functions to obtain a numerical verification of the infinite dimensional inequalities, see for instance [17], where an orthogonal set of polynomials were used, and [40], where a standard polynomial basis is used. For timedelay systems, a particular class of infinite-dimensional systems, necessity and sufficiency of a class of LFs, called complete quadratic functionals, has been shown in [22,Chapter 5].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

Gahlawat

Valmórbida

2020

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

We present a method for the stability analysis of a large class of linear Partial Differential Equations (PDEs) in one spatial dimension. We rely on Lyapunov analysis to establish the exponential stability of the systems under consideration. The proposed test for the verification of the underlying Lyapunov inequalities relies on the existence of solutions of a system of coupled differential equations. We illustrate the application of this method using a PDE actuated by a backstepping computed feedback law. Furthermore, for the case of PDEs defined by polynomial data, we formulate a numerical methodology in the form of a convex optimization problem which can be solved algorithmically. We show the effectiveness of the proposed numerical methodology using examples of different types of PDEs.

show abstract

“…with V d (u) defined as in (17), S(u) defined as in (30), and R (u) defined as in (24). Therefore, we may use (38) and (39) to conclude…”

Section: Resultsmentioning

confidence: 99%

“…where V and V d are defined in (16) and (17), respectively. Then, there exists a positive scalar κ such that the solutions of (8) satisfy…”

Section: Problem Statementmentioning

confidence: 99%

“…where h1 is defined in (50). Using these functions and the definitions in (46), we may apply (17) to compute…”

Section: A Boundary Controlled Parabolic Pde-backstepping Feedback Lawmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

Gahlawat

Valmórbida

2020

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

show abstract

“…The details of the following analysis are quite technical, and will be omitted to keep the focus on our main objective -formulating an SDP whose solution gives a feasible background field for (29). LMI relaxations of integral inequalities using Legendre series will be thoroughly discussed in a future publication [24]. Moreover, for notational neatness, we will drop the suffix m from Q m , W m and α m ; it should be understood that the following analysis holds for each individual m ≥ 1.…”

Section: Rigorous Finite Dimensional Relaxation Of Qm Using Legendmentioning

confidence: 99%

Optimal bounds with semidefinite programming: An application to stress-driven shear flows

Fantuzzi

Wynn

2016

Phys. Rev. E

View full text Add to dashboard Cite

We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler-Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times, and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface. Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress.

show abstract

The background method: theory and computations

Fantuzzi

Arslan

Wynn

2022

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader ‘auxiliary function’ framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh–Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.

show abstract

Semidefinite relaxation of a class of quadratic integral inequalities

Cited by 5 publications

References 20 publications

Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

Optimal bounds with semidefinite programming: An application to stress-driven shear flows

The background method: theory and computations

Contact Info

Product

Resources

About