On a Class of Second-Order PDE&amp;PDI Constrained Robust Modified Optimization Problems

Savin Treanţă

doi:10.3390/math9131473

Abstract

In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Concretely, we introduce and investigate an auxiliary (modified) variational control problem, which is much easier to study, and provide some equivalence results by using the notion of a normal weak robust optimal solution.

Highlights

Over time, several researchers have taken a particular interest in studying some optimization problems with ODE, partial differential equations (PDEs), or isoperimetric constraints
We introduce the notion of a weak robust optimal solution for the considered class of constrained optimization problems
Let us proceed by contradiction and consider ( ā, c) is not a weak robust optimal solution to the modified multi-dimensional scalar optimization problem ( P)(ā,c)

Summary

Introduction

The partial differential equations (PDEs) and inequations (PDIs) are very important in the study of many processes and phenomena in nature, science and engineering In this regard, over time, several researchers have taken a particular interest in studying some optimization problems with ODE, PDE, or isoperimetric constraints. Many researchers turned their attention to real problems involving higher-order PDEs, isoperimetric restrictions, uncertain data, or a combination thereof In this respect, the reader is directed to the following research works: Liu and Yuan [10], Jeyakumar et al [11], Wei et al [12], Preeti et al [13], Sun et al [14], Treanţă [15], Lu et al [16].

Preliminary Tools

The Associated Modified Optimization Problem

Findings

Conclusions