12,497 research outputs found
An applied mathematical excursion through Lyapunov inequalities, classical analysis and differential equations
Several different problems make the study of the so called Lyapunov type
inequalities of great interest, both in pure and applied mathematics. Although
the original historical motivation was the study of the stability properties of
the Hill equation (which applies to many problems in physics and engineering),
other questions that arise in systems at resonance, crystallography,
isoperimetric problems, Rayleigh type quotients, etc. lead to the study of
Lyapunov inequalities () for differential equations.
In this work we review some recent results on these kinds of questions which
can be formulated as optimal control problems. In the case of Ordinary
Differential Equations, we consider periodic and antiperiodic boundary
conditions at higher eigenvalues and by using a more accurate version of the
Sturm separation theory, an explicit optimal result is obtained. Then, we
establish Lyapunov inequalities for systems of equations. To this respect, a
key point is the characterization of the best Lyapunov constant for the
scalar given problem, as a minimum of some especial (constrained or
unconstrained) variational problems defined in appropriate subsets of the usual
Sobolev spaces. For Partial Differential Equations on a domain , it is proved that the relation between the quantities and
plays a crucial role in order to obtain nontrivial Lyapunov type
inequalities (which are called Sobolev inequalities by many authors). This fact
shows a deep difference with respect to the ordinary case. Combining the linear
results with Schauder fixed point theorem, we can obtain some new results about
the existence and uniqueness of solutions for resonant nonlinear problems for
ODE or PDE, both in the scalar case and in the case of systems of equationsComment: 36 page
Stability of the solution set of quasi-variational inequalities and optimal control
For a class of quasi-variational inequalities (QVIs) of obstacle-type the
stability of its solution set and associated optimal control problems are
considered. These optimal control problems are non-standard in the sense that
they involve an objective with set-valued arguments. The approach to study the
solution stability is based on perturbations of minimal and maximal elements of
the solution set of the QVI with respect to {monotone} perturbations of the
forcing term. It is shown that different assumptions are required for studying
decreasing and increasing perturbations and that the optimization problem of
interest is well-posed.Comment: 29 page
Integral Inequalities and their Applications to the Calculus of Variations on Time Scales
We discuss the use of inequalities to obtain the solution of certain
variational problems on time scales.Comment: To appear in Mathematical Inequalities & Applications
(http://mia.ele-math.com). Accepted: 14.01.201
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