Quadratic Integral Inequalities Research Articles

A second order quadratic integral inequality associated with regular problems

<abstract><p>In this paper, we establish a quadratic integral inequality involving the second order derivative of functions in the following form: for all $ f\in D $,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray} \int_{a}^{b}{r|f''|^{2}+p|f'|^{2}+q|f|^{2}}\geq\mu_{0}\int_{a}^{b}|f|^{2}. \end{eqnarray} $\end{document} </tex-math></disp-formula></p> <p>Here $ r, p, q $ are real- valued coefficient functions on the compact interval $ [a, b] $ with $ r(x) > 0 $. $ D $ is a linear manifold in the Hilbert function space $ L^{2}(a, b) $ such that all integrals of the above inequality are finite and $ \mu_{0} $ is a real number that can be determined in terms of the spectrum of a uniquely determined self adjoint differential operator in $ L^{2}(a, b) $. The inequality is the best possible, i.e., the number $ \mu_{0} $ cannot be increased. $ f $ is a complex-valued function in $ D. $</p></abstract>

A second order quadratic integral inequality with an application to ordinary differential operators

A second order quadratic integral inequality with an application to ordinary differential operators

Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints

We introduce a new technique to optimize a linear cost function subject to a one-dimensional affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in stability analysis, input-to-state/output analysis, and control of many systems governed by partial differential equations (PDEs), in particular fluid dynamical systems. First, we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that under mild assumptions a convergent, non-decreasing sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can also be computed with SDPs. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our techniques by solving problems arising from the stability analysis of PDEs.

On Extremal Properties of Boundary Points of Reachable Sets for a System with Integrally Constrained Control

We consider a nonlinear affine-control system with constraints on the control variables given by a quadratic integral inequality. It is shown that any admissible control that steers the control system to the boundary of its reachable set is a local solution of an optimal control problem. A numerical procedure for calculating the reachable sets based on the Pontryagin maximum principle is proposed.

On some further second order integral inequalities

Some weighted quadratic integral inequalities of the second order involving a function, its first and second derivative has been derived using the uniform method of obtaining integral inequalities. An example in which some new integral inequalities with Chebyshev weight functions appeared has been considered. Mathematics subject classification (2000): 26D10.

On quadratic integral inequalities of the second order

The aim of this paper is to generalize the uniform method of obtaining integral inequalities in order to derive inequalities involving a function h, its first and second derivatives with weights. Such inequalities have been considered before by others, but other methods were applied. Our method makes it possible to obtain, in a natural way, the equality conditions important in differential equations. Moreover it allows us to avoid some assumptions on weights that have to be given in other methods. Then the inequality will be examined in order to simplify the boundary conditions for h. These considerations will be followed by examples with Chebyshev weight functions and constant weights with the classical Hardy, Littlewood, Polya inequality as a special case.

Second Variation Condition and Quadratic Integral Inequalities with Higher Order Derivatives

Second Variation Condition and Quadratic Integral Inequalities with Higher Order Derivatives

Some second-order integral inequalities of generalized Hardy type

A uniform method of obtaining various types of integral inequalities involving a function and its first derivative is extended to integral inequalities involving a function and its second derivative. Specifically, some quadratic integral inequalities of generalized Hardy type involving a function and its second derivative are derived and examined. The functions for which the inequalities hold are characterized by boundary conditions.

Some quadratic integral inequalities of first order

Some quadratic integral inequalities of first order

Some quadratic integral inequalities of Opial type

Some quadratic integral inequalities of Opial type

On a quadratic integral inequality

SynopsisThe inequality considered iswhere p and q are given real-valued coefficients on the interval [a, b), with b ≦ ∝, of the real line; here D is a linear manifold of the Hilbert function space L2(a, b), and μ is a real number characterised in terms of the spectrum of a uniquely determined self-adjoint differential operator in L2(a, b).

Growth and Continuity of Functions Satisfying Quadratic Integral Inequalities

Growth and Continuity of Functions Satisfying Quadratic Integral Inequalities