Convergence of difference approximations and regularization of the optimal control problem for an ordinary linear differential equation of the second order

In a recent paper (1) I studied a class of generalized convex functions of a single real variable which I called sub-(L) functions. Given an ordinary linear differential equation of the second order L(y) = 0, a function f(x) is sub-(L) in (a, b) if it is majorized there by the solutions of the equation. More precisely, for every x1, x2 in (a, b),f(x) ≤ F12(x) in (x1, x2), where F12 is that solution of L(y) = 0 (supposed unique) which takes the values f(xi) at xi. It was found that sub-(L) functions are characterized in a manner closely analogous to ordinary convex functions.

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Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Analysis ◽

10.1515/anly-2019-0049 ◽

2020 ◽

Vol 40 (3) ◽

pp. 127-150

Author(s):

Tania Biswas ◽

Sheetal Dharmatti ◽

Manil T. Mohan

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Control Problem ◽

Stokes Equations ◽

Minimum Principle ◽

Immiscible Fluids ◽

Second Order ◽

Navier Stokes ◽

Distributed Optimal Control ◽

Navier Stokes Equations

AbstractIn this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

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An oscillation criterion for a forced second order linear differential equation

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1993-1154243-9 ◽

1993 ◽

Vol 118 (3) ◽

pp. 813-813 ◽

Cited By ~ 1

Author(s):

M. A. El-Sayed

Keyword(s):

Differential Equation ◽

Linear Differential Equation ◽

Second Order ◽

Oscillation Criterion ◽

Order Linear Differential Equation ◽

Order Linear ◽

Linear Differential

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A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026330 ◽

1986 ◽

Vol 102 (3-4) ◽

pp. 253-257 ◽

Cited By ~ 4

Author(s):

B. J. Harris

Keyword(s):

Differential Equation ◽

Asymptotic Expansion ◽

Linear Differential Equation ◽

Asymptotic Series ◽

Second Order ◽

Order Linear Differential Equation ◽

Order Linear ◽

The Real ◽

Linear Differential ◽

Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

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