scholarly journals On Well-Posedness of Some Constrained Variational Problems

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2478
Author(s):  
Savin Treanţă
Keyword(s):  
Variational Inequality ◽  
Lower Semicontinuity ◽  
Integral Functional ◽  
Inequality Constraints ◽  
Multiple Integral ◽  
Variational Problems ◽  
Well Posedness ◽  
New Class ◽  
Scalar Multiple ◽  
Constrained Variational Problems

By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, in this paper we study the well-posedness of a new class of variational problems with variational inequality constraints. More specifically, by defining the set of approximating solutions for the class of variational problems under study, we establish several results on well-posedness.

scholarly journals Well Posedness of New Optimization Problems with Variational Inequality Constraints

2021 ◽  
Vol 5 (3) ◽  
pp. 123
Author(s):  
Savin Treanţă
Keyword(s):  
Variational Inequality ◽  
Lower Semicontinuity ◽  
Optimization Problems ◽  
Integral Functional ◽  
Inequality Constraints ◽  
Multiple Integral ◽  
Well Posedness ◽  
Constrained Optimization Problems ◽  
New Class ◽  
Theoretical Developments

In this paper, we studied the well posedness for a new class of optimization problems with variational inequality constraints involving second-order partial derivatives. More precisely, by using the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity for a multiple integral functional, and by introducing the set of approximating solutions for the considered class of constrained optimization problems, we established some characterization results on well posedness. Furthermore, to illustrate the theoretical developments included in this paper, we present some examples.

scholarly journals Duality Theorems for (ρ,ψ,d)-Quasiinvex Multiobjective Optimization Problems with Interval-Valued Components

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 894
Author(s):  
Savin Treanţă
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Integral Functional ◽  
Multiple Integral ◽  
Control Problems ◽  
Duality Theorems ◽  
New Class ◽  
Duality Results ◽  
Multiobjective Optimization Problems ◽  
Interval Valued

The present paper deals with a duality study associated with a new class of multiobjective optimization problems that include the interval-valued components of the ratio vector. More precisely, by using the new notion of (ρ,ψ,d)-quasiinvexity associated with an interval-valued multiple-integral functional, we formulate and prove weak, strong, and converse duality results for the considered class of variational control problems.

scholarly journals On a Class of Second-Order PDE&PDI Constrained Robust Modified Optimization Problems

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1473
Author(s):  
Savin Treanţă
Keyword(s):  
Optimization Problems ◽  
Integral Functional ◽  
Optimal Solution ◽  
Multiple Integral ◽  
Second Order ◽  
Control Problems ◽  
Mathematical Framework ◽  
Mixed Constraints ◽  
Scalar Multiple ◽  
Cost Functionals

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.

scholarly journals Upper and Lower bounds for a certain class of constrained variational problems

1963 ◽  
Vol 3 (4) ◽  
pp. 449-453
Author(s):  
M. A. Hanson
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Existence Of Solutions ◽  
Necessary Conditions ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Variational Problems ◽  
Upper And Lower Bounds ◽  
Control Problems ◽  
Constrained Variational Problems

Certain optimization problems involving inequality constraints, known as optimal control problems have been extensively studied during recent years especially in relation to the calculation of optimal rocket thrusts and trajectories. A summary of these works is given by Berkovitz [1] who also establishes necessary conditions for the existence of solutions for a wide class of such problems.

scholarly journals On well-posedness associated with a class of controlled variational inequalities

2021 ◽  
Author(s):  
Savin Treanta ◽  
Shalini Jha
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Integral Functional ◽  
Solution Set ◽  
Variational Inequality Problems ◽  
Well Posedness ◽  
New Concepts ◽  
Curvilinear Integral ◽  
Theoretical Developments

In this paper, by using the new concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with the considered curvilinear integral functional, we investigate the well-posedness and well-posedness in generalized sense for a class of controlled variational inequality problems. More precisely, by introducing the approximating solution set of the considered class of controlled variational inequality problems, we formulate and prove some characterization results on well-posedness and well-posedness in generalized sense. Also, the theoretical developments presented in the paper are accompanied by illustrative examples.

scholarly journals On a Class of Differential Variational Inequalities in Infinite-Dimensional Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 266 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Optimal Control Theory ◽  
Infinite Dimensional ◽  
Nash Games ◽  
New Class ◽  
Set Valued Mappings ◽  
Differential Variational Inequalities ◽  
Theoretical Developments

A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in infinite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games.

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