Linear Operator Inequalities Research Articles

On Numerical Radius Inequalities for Hilbert Space Operators

This paper aims to find special cases of some inequalities for numerical radii and spectral radii of a bounded linear operator on a Hil-bert space, we focus on numerical radii inequalities for restricted linear operators on complex Hil-bert spaces for the case of one and two operators, and study the numerical range of an operator K on a complex Hil-bert space H, after that we present some inequalities for numerical radii and spectral radii and studied it to find new results. At the end of this paper we find several inequalities for numerical radii by using the spectral norm, this study is necessary to find other bound for zeros of polynomials and this study is necessary to find new bounds for the zeros of polynomials by a playing the new results to the companion matrix.

Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables where the disturbance does not enter through the boundary.For such systems, we define a notion of H2-norm using an initial state-to-output framework and show that this notion reduces to more traditional concepts under the assumption of existence of a strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE representation and for such systems show that computing a minimal upper bound on the H2-norm of delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation results are available in the literature. The numerical results demonstrate the accuracy of the computed upper bound on the H2-norm.

Delay-Dependent Stability for Load Frequency Control System via Linear Operator Inequality

The stability analysis problem is considered for multiarea load frequency control (LFC) systems with electric vehicles (EVs) and time delays. The novel linear operator inequality approach is proposed and a less conservative delay-dependent stability condition is achieved. First, the model of the multiarea LFC system with EVs and delays is expressed by the partial integral equation (PIE) at the first time. Then, a complete quadratic Lyapunov-Krasovskii functional is built in the form of an inner product of the linear partial integral (PI) operator. The novel stability criteria with less conservatism are proposed in the form of linear operator inequality. Moreover, the relationships between the delay margins and the controller parameters are shown. Finally, the simulations are conducted on both one-area and two-area LFC systems to show the effectiveness of the proposed approach.

A Convex Solution of the $H_\infty$-Optimal Controller Synthesis Problem for Multidelay Systems

Optimal controller synthesis is a bilinear problem and hence difficult to solve in a computationally efficient manner. We are able to resolve this bilinearity for systems with delay by first convexifying the problem in infinite dimensions, i.e., formulating the $H_\infty$-optimal state-feedback controller synthesis problem for distributed-parameter systems as a linear operator inequality, which a form of convex optimization with operator variables. Next, we use positive matrices to parameterize positive “complete quadratic” operators, allowing the controller synthesis problem to be solved using semidefinite programming (SDP). We then use the solution to this SDP to calculate the feedback gains and provide effective methods for real-time implementation. Finally, we use several test cases to verify that the resulting controllers are optimal to several decimal places as measured by the minimal achievable closed-loop $H_\infty$ norm, and as compared against controllers designed using high-order Padé approximations.

Improved Young's inequalities for positive linear operators

Improved Young's inequalities for positive linear operators

Improved Young's inequalities for positive linear operators

Improved Young's inequalities for positive linear operators

On families of optimal Hardy-weights for linear second-order elliptic operators

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a,b), ∞≤a<b≤∞, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.

Exponential Inequalities for Positive Linear Mappings

In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pečarić method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means inequalities.

Extrapolation in the scale of generalized reverse Hölder weights

We develop a theory of extrapolation for weights that satisfy a generalized reverse Holder inequality in the scale of Orlicz spaces. This extends previous results by Auscher and Martell (Adv Math 212(1):225–276, 2007) on limited range extrapolation. We then provide several applications of our extrapolation techniques. These applications include new results and new proofs of known results for two weight inequalities for linear and bilinear operators.

Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set DL, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) L(X)=A0⊗I+∑j=1gAj⊗Xj⪰0, where Aj are self-adjoint linear operators on a separable Hilbert space, Xj matrices and I is an identity matrix. If Aj are matrices, then L(X)⪰0 is called a linear matrix inequality (LMI) and DL a free spectrahedron. For monic LMIs, i.e., A0=I, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion DL1⊆DL2 from monic LMIs to monic LOIsL1 and L2. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron DL and its projection, called a free Hilbert spectrahedrop. Further on, using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials F positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fejér–Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials F. Finally, focusing on the algebraic description of the equality DL1=DL2, we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs case by counterexamples.

Reverses of Young and Heinz inequalities for positive linear operators

Let A, B be invertible positive operators on a Hilbert space H. We present some improved reverses of Young type inequalities, in particular, $$ (1-\nu)^{2\nu}(A\nabla B)+(1-\nu)^{2(1-\nu)}H_{2\nu}(A,B) \geq2(1-\nu ) (A\sharp B) $$ and $$ (1-\nu)^{2\nu}H_{2\nu}(A,B)+(1-\nu)^{2(1-\nu)}(A\nabla B) \geq2(1-\nu ) (A\sharp B), $$ where $0\leq\upsilon\leq\frac{1}{2}$ . We also give some new inequalities involving the Heinz mean for the Hilbert-Schmidt norm.

Dissipativity-based observer design for a class of coupled 1-D semi-linear parabolic PDE systems

In the present paper the dissipativity-based observer design approach is extended to a class of 1-D semi-linear parabolic PDE systems with pointwise in-domain measurement. On the basis of a modal innovation over the set of slow eigenvalues of the linear operator and the consideration of a dissipativity (sector) condition for the nonlinearity an exponential convergence condition for the estimation error is derived in form of a linear operator inequality, which is solved on the basis of an eigenfunction expansion of the linear operator. The approach is illustrated by numerical simulations for a coupled two-state semi-linear PDE system.

Control synthesis of linear distributed parameter switched systems

The control synthesis for switched systems is extended to distributed parameter switched systems in Hilbert space. Based on semigroup and operator theory, by means of multiple Lyapunov method incorporated average dwell time approach, sufficient conditions are derived in terms of linear operator inequalities framework for distributed parameter switched systems. Being applied to one dimensional heat propagation switched systems, these linear operator inequalities are reduced to linear matrix inequalities subsequently. In particular, the state feedback gain matrices and the switching law are designed, and the state decay estimate is explicitly given whose decay coefficient completely depends on the system's parameter and the boundary condition. Finally, two numerical examples are given to illustrate the proposed method.

Design of a Proportional Integral Control Using Operator Theory for Infinite Dimensional Hyperbolic Systems

This brief considers the control design of a nonlinear distributed parameter system in infinite dimension, described by the hyperbolic partial differential equations of de Saint-Venant. The nonlinear system dynamic is formulated by a multimodels approach over a wide operating range, where each local model is defined around a set of operating regimes. A new proportional integral feedback is designed and performed through bilinear operator inequality and linear operator inequality techniques for infinite dimensional systems. The new results have been simulated and also compared with previous results in finite and infinite dimension, to illustrate the new theoretical contribution.

Exponential stability of linear distributed parameter switched systems with time-delay

In this paper, the exponential stability analysis for ODE switched systems with time delay is extended to distributed parameter switched systems (DPSS) in Hilbert space. For a given family of exponential stable subsystems, this paper focuses on finding conditions to guarantee the overall DPSS’ exponential stability. Based on semigroup theory, by applying piecewise Lyapunov-Krasovskii functionals method incorporated average dwell time approach, sufficient conditions for exponential stability are derived. These conditions are given in the form of linear operator inequalities (LOIs) where the decision variables are operators in Hilbert space, and the stability properties depend on switching rule. Being applied to heat switched propagation equations, these LOIs are reduced to standard Linear Matrix Inequalities (LMIs). Finally, a numerical example is given to illustrate the effectiveness of the proposed result.

Linear Operator Inequality and Null Controllability with Vanishing Energy for Unbounded Control Systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $ y_0 \in H$ we associate the minimal “energy” needed to transfer $ y_0 $ to $ 0 $ in a time $ T \ge T_0$ (``energy” of a control being the square of its $ L^2 $ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk [SIAM J. Control Optim., 42 (2003), pp. 1013--1032] for distributed systems. The proofs in the Priola--Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the linear operator inequality.

Input-to-State Stability of Lur’e Hyperbolic Distributed Complex-Valued Parameter Control Systems: LOI Approach

In this work, input-to-state stability of Lur’e hyperbolic distributedcomplex-valuedparameter control systems has been addressed. Using comparison principle, delay-dependent sufficient conditions for the input-to-state stability in complex Hilbert spaces are established in terms of linear operator inequalities. Finally, numerical computation illustrates our result.

Weighted Norm Inequalities for Convolutions, Differential Operators, and Generalized Hypergeometric Functions

The recent work on discrete inequalities with the binomially made weights leads to a sharp weighted norm inequality for convolutions of complex-valued functions on a finite interval. This result and the Hadamard-type modifications of power series provide the basis for an integral approach to solution-kernel estimates for Volterra convolution equations and for various integro-differential and special function applications. The development of this approach and the new weighted norm inequalities for linear differential operators and generalized hypergeometric functions are presented in this paper.

Outils d’information/prévention sur les risques professionnels dans la filière Bois

In this work, dissipativity of Lur’e distributed parameter control systems has been addressed. Delay-dependent sufficient conditions for the dissipativity with respect to the infinite-dimensional version of energy supply rate (Q1,S1,R1) characterized exclusively by unbounded operator Q1 are established in terms of linear operator inequalities (LOIs). Finally, the heat equation illustrates our result.

Exponential stability of non-linear hyperbolic distributed complex-valued parameter systems: The linear fuzzy operator inequality approach

In this work, exponential stability of non-linear hyperbolic distributed complex-valued parameter systems has been addressed. Using a linear fuzzy operator inequality approach, which is a novel notion proposed for the first time in this work, delay-dependent sufficient conditions for the exponential stability in complex Hilbert spaces are established in terms of linear matrix inequalities (LMIs). Finally, numerical computation illustrates our result.