Variational Control Problems Research Articles

Isoperimetric-type constrained variational control problem with uncertainty: robust optimality and duality

Isoperimetric-type constrained variational control problem with uncertainty: robust optimality and duality

On some connections between interval-valued variational control problems and the associated inequalities

The principal purpose of this paper is to investigate the relationships between a class of interval-valued optimization problems and the associated inequalities. Concretely, by using the solutions of the corresponding (weak, split) inequalities, we establish some existence results of (strong) LU-optimal solutions for the considered interval-valued optimization problem. In this regard, the LU-convexity property of integral functionals plays an important role. Also, an illustrative application is also presented that verifies the statements and the theoretical results.

Efficiency in vector ratio variational control problems involving geodesic quasiinvex multiple integral functionals

Efficiency in vector ratio variational control problems involving geodesic quasiinvex multiple integral functionals

On Sufficiency Conditions for Some Robust Variational Control Problems

We study the sufficient optimality conditions for a class of fractional variational control problems involving data uncertainty in the cost functional. Concretely, by using the parametric technique, we prove the sufficiency of the robust necessary optimality conditions by considering convexity, quasi-convexity, strictly quasi-convexity, and/or monotonic quasi-convexity assumptions of the involved functionals.

Variational problem of adaptive optimal control. Theoretical and applied computer analysis

The problem of adaptive optimal control of a dynamical system, which belongs to the class of conditional variational problems with moving boundaries, is considered. A variational and computer study of the controlled adaptive motion of a material point is carried out for the problem of the energy quality functional minimizing with a moving, not predetermined right transboundary and in the case when the mass of the point changes depending on the unfixed final time. The problem is solved using the schemes and procedures of the classical calculus of variations, as well as adaptive estimation techniques, including the derivation of the variation of the auxiliary quality functional, the corresponding Euler equations, and the adaptive estimation algorithm. When solving a general conditional variational problem, the obtained closed system of differential equations was studied for the formation of an adaptive optimal control system for a dynamic plant with a given performance functional. The results of the unconditional formulation of the problem are generalized to the case of additional differential (nonholonomic) and holonomic constraints. In a variational adaptive optimal control problem, the transversality condition is formulated in terms of the local programming condition. The developed variational scheme of adaptive optimal synthesis can be used in the calculation and design of controlled dynamic systems. This optimization scheme is also promising for use in systems where operating time is non-fixed in advance. The results achieved in this paper concern obtaining specific equations, expressions, and formulas relative to the model example under study and finding graphs of the main time functions that determine the nature of the movement of the control object and the quality of the corresponding transients. The proposed adaptive optimal control algorithms for purposeful movement of the studied material point were tested in digital mode and showed their effectiveness which makes them promising for further use in more complex nonlinear adaptive systems of dynamic optimal control.

On a minimal criterion of efficiency in vector variational control problems

In this paper, we establish a minimal criterion of efficiency for a class of multiobjective variational control problems. Thus, some minimal conditions are formulated so that all local efficient solutions are also global. Also, we present an example to illustrate the mathematical developments derived in the paper.

On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems

In this paper, we consider the class of nondifferentiable multiobjective fractional variational control problems involving the nondifferentiable terms in the numerators and in the denominators. Under univexity and generalized univexity hypotheses, we prove optimality conditions and various duality results for such nondifferentiable multiobjective fractional variational control problems. The results established in the paper generalize many similar results established earlier in the literature for such nondifferentiable multiobjective fractional variational control problems.

Robust Efficiency Conditions in Multiple-Objective Fractional Variational Control Problems

The aim of this study is to investigate multi-dimensional vector variational problems considering data uncertainty in each of the objective functional and constraints. We establish the robust necessary and sufficient efficiency conditions such that any robust feasible solution could be the robust weakly efficient solution for the problems under consideration. Emphatically, we present robust efficiency conditions for multi-dimensional scalar, vector, and vector fractional variational problems by using the notion of a convex functional.

On some variational inequality-constrained control problems

In this paper, by considering some properties associated with scalar functionals of multiple-integral type, we study the well-posedness and generalized well-posedness for a new variational inequality-constrained optimization problems By using the set of approximating solutions, we state some characterization theorems on well-posedness and generalized well-posedness. Also, in order to validate the derived results, some examples are given.

LU-Optimality Conditions in Optimization Problems With Mechanical Work Objective Functionals.

In this article, we introduce interval-valued Kuhn-Tucker (KT)-pseudoinvex optimization problems governed by interval-valued path-independent curvilinear integral objective functionals. Concretely, it is proven that an interval-valued KT-pseudoinvex variational control problem is described such that every KT point is an LU-optimal solution. In addition, the main results are highlighted by two illustrative applications describing the controlled behavior of an artificial neural system.

Necessary and sufficient optimality conditions for some robust variational problems

AbstractIn this article, we complete the study started in Treanţă on multidimensional scalar variational control problems with mixed constraints implying partial differential equations and inequations of second‐order. More specifically, under convexity assumptions of the involved uncertain functionals of multiple integral type, we state necessary and sufficient optimality conditions for a class of robust variational problems with uncertainty in the constraint and objective functionals.

Recent Advances of Constrained Variational Problems Involving Second-Order Partial Derivatives: A Review

This paper comprehensively reviews the nonlinear dynamics given by some classes of constrained control problems which involve second-order partial derivatives. Specifically, necessary optimality conditions are formulated and proved for the considered variational control problems governed by integral functionals. In addition, the well-posedness and the associated variational inequalities are considered in the present review paper.

A Modified Numerical Method Based on Bernstein Wavelets for Numerical Assessment of Fractional Variational and Optimal Control Problems

A Modified Numerical Method Based on Bernstein Wavelets for Numerical Assessment of Fractional Variational and Optimal Control Problems

Constrained controlled optimization problems involving second-order derivatives

In this paper, we investigate the nonlinear Lagrange dynamics associated with two classes of constrained controlled optimization problems involving second- order derivatives. More precisely, we formulate and prove necessary conditions of optimality for the considered variational control problems governed by simple integral functionals and second-order ordinary differential equation (ODE)/ isoperimetric constraints. Moreover, we propose an algorithm to synthesize the concrete steps to be followed for solving constrained controlled optimization problems.

Numerical solution of fractional variational and optimal control problems via fractional-order Chelyshkov functions

<abstract><p>In this paper, we present a new numerical method based on the fractional-order Chelyshkov functions (FCHFs) for solving fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). The fractional derivatives are considered in the Caputo sense. The operational matrix of fractional integral for FCHFs, together with the Lagrange multiplier method, are used to reduce the fractional optimization problem into a system of algebraic equations. Some results concerning the approximation errors are discussed and the convergence of the presented method is also demonstrated. The performance of the introduced method is tested through several examples. Some comparisons with recent numerical methods are introduced to show the accuracy and effectiveness of the presented method.</p></abstract>

Penalty approach for KT-pseudoinvex multidimensional variational control problems

<abstract><p>The present paper is the result of a contemplative study of a multi-time control problem (MCP) by considering its associated equivalent auxiliary control problem (MCP)$ _{\varsigma} $ via the exact $ l_{1} $ penalty method. Further study reveals that the solution set of the considered problem and the auxiliary problem exhibits an equivalence under the KT-pseudoinvexity hypothesis. Moreover, the study is extended towards the saddle point defined for (MCP) to establish the relationship between the solution set of multi-time control problem (MCP) and its associated equivalent auxiliary control problem (MCP)$ _{\varsigma} $. Finally, we present an illustrative application to authenticate the results presented in this paper.</p></abstract>

On well-posed isoperimetric-type constrained variational control problems

In this paper, we investigate well-posedness and well-posedness in the generalized sense for a new class of isoperimetric-type constrained variational control problems. For this purpose, in the first part of the current paper, we introduce new versions for the concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with the considered curvilinear integral functional. Thereafter, we define the approximating solution set of the considered class of isoperimetric-type constrained variational control problems. By using these completely new elements, we formulate and prove several characterization results on well-posedness and well-posedness in the generalized sense for the problem under study. Also, in order to highlight the theoretical results and tools derived in the paper, some illustrative examples are provided.

Robust saddle‐point criterion in second‐order partial differential equation and partial differential inequation constrained control problems

AbstractIn this article, for a given class of multi‐dimensional scalar variational control problems (named ) with mixed constraints implying second‐order partial differential equations and inequations, we introduce an auxiliary (modified) class of variational control problems (named ), which is much easier to study, and provide some characterization results of and by using the notions of normal weak robust optimal solution and robust saddle‐point associated with a Lagrange functional corresponding to . For this aim, we consider 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.

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

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.

On a Class of Isoperimetric Constrained Controlled Optimization Problems

In this paper, we investigate the Lagrange dynamics generated by a class of isoperimetric constrained controlled optimization problems involving second-order partial derivatives and boundary conditions. More precisely, we derive necessary optimality conditions for the considered class of variational control problems governed by path-independent curvilinear integral functionals. Moreover, the theoretical results presented in the paper are accompanied by an illustrative example. Furthermore, an algorithm is proposed to emphasize the steps to be followed to solve a control problem such as the one studied in this paper.