Multidimensional Variational Problems Research Articles

On convergence of exponential penalty for the multi-dimensional variational problems

In this article, we describe a method to deal with a multi-dimensional variational problem with inequality constraints using an exponential penalty function. We formulate an unconstrained multi-dimensional variational problem and examine the relationships between the optimal solution to the considered multi-dimensional variational problem and the sequence of minimizers of the unconstrained multi-dimensional variational problem. The convergence of the proposed exponential penalty approach is also investigated, which shows that a convergent subsequence of the sequence of minimizers of the unconstrained multi-dimensional variational problem approaches an optimal solution to the multi-dimensional variational problem. Further, an illustrative application (to minimize a manufacturing cost functional of a production firm) is also presented to confirm the effectiveness of the proposed outcomes.

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 a global efficiency criterion in multiobjective variational control problems with path-independent curvilinear integral cost functionals

In this paper, a global efficiency criterion is established for a class of multidimensional variational control problems governed by first order PDE&PDI constraints and path-independent curvilinear integral cost functionals. More precisely, a minimal efficiency criterion for a local efficient solution to be its global efficient solution in the considered optimization problem is formulated and proved. Also, the theoretical developments derived in the paper are accompanied by an example of a nonconvex optimization problem.

Geometrodynamic Models of Continuum Mesomechanics: Dynamic Degrees of Freedom with Non-Eulerian Space-Time Evolution

The paper proposes a Lagrangian formalism for describing the space-time dynamics of complex continuum motion with addition variables—non-Eulerian dynamic degrees of freedom. The new variables are interpreted in terms of mechanics and geometry, and principles are suggested for their tracking in experiments. The formalism offers invariant tensor representations of Lagrangians in a system with an extended set of independent variables, explains the mechanical meaning of their respective coefficients, and gives Euler—Lagrange equations for this type of multidimensional variational problems. It is hypothesized that the bundle geometry of dynamic degrees of freedom is a generalized structure for fluctuation and other models of complex continuum evolution. The proposed method is analyzed as applied to dynamic equations of developed turbulence, and an interpretation on its basis is given to turbulence degeneration into unsteady Euler fields.

Isoperimetric and Functional Inequalities

We establish lower estimates for an integral functional$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$where \(\Omega\) -- a bounded domain in \(\mathbb{R}^n \; (n \geqslant 2)\), an integrand \(f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)\) -- a function that is \(B\)-measurable with respect to a variable \(t\) and is convex and even in the variable \(p\), \(\nabla u(x)\) -- a gradient (in the sense of Sobolev) of the function \(u \colon \Omega \rightarrow \mathbb{R}\). In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality \(H^{n-1}( \partial A) \geqslant \lambda(m_n A)\), that connects \((n-1)\)-dimensional Hausdorff measure \(H^{n-1}(\partial A )\) of relative boundary \(\partial A\) of the set \(A \subset \Omega\) with its \(n\)-dimensional Lebesgue measure \(m_n A\). The integrand \(f\) is assumed to be isotropic, i.e. \(f(t,p) = f(t,q)\) if \(|p| = |q|\).Applications of the established results to multidimensional variational problems are outlined. For functions \( u \) that vanish on the boundary of the domain \(\Omega\), the assumption of the isotropy of the integrand \( f \) can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand \( f \) and of the function \( u \). The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function \(u\), but also to the integrand \(f\). The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.

Quantum theory of particles and fields as an extension of a probabilistic variational approach to classical mechanics and classical field theory: I

A theoretical scheme, based on a probabilistic generalization of Hamilton's principle, is elaborated to obtain a unified description of more general dynamical behaviors determined both from a Lagrangian function and by mechanisms not contemplated by this function. Within this scheme, quantum mechanics, classical field theory and a quantum theory for scalar fields are discussed. As a by-product of the probabilistic scheme for classical field theory, the equations of the De Donder–Weyl theory for multi-dimensional variational problems are recovered.

An application of the algorithm of the cyclic coordinate descent in multidimensional optimization problems with constrained speed

In this paper we present an application of the algorithm of the cyclic coordinate descent in multidimensional variational problems with constrained speed, the physical motivation of the problem being the optimization of hydrothermal systems. The proof of the convergence of the succession generated by the algorithm was based on the use of an appropriate adaptation of Zangwill’s global theorem of convergence. We have also included an algorithm for the formal construction of the descending succession (the solution of an optimum control problem), the approximation of which we carried out using an adaptation of the Euler method in conjunction with a procedure inspired by the shooting method.

Relaxation of multidimensional variational problems with constraints of general form

This paper is devoted to further developement of an idea of a well-known theorem of Bogolubov [2]. Here we construct a relaxation of multidimensional variational problems with constraints of rather general form on gradients of admissible functions; it is assumed that the gradient of an admissible function belongs to an arbitrary bounded set. This relaxation involves as a class of admissible functions the closure of the class of admissible functions of the original problem in the topology of uniform convergence, and uses a theorem characterizing this closure, which is proved in [15]. The case when the gradient of an admissible function is constrained within a bounded closed convex body is studied in the works [13,15,19]. The study of multidimensional variational problems was started in 1970s by Ekeland and Temam [13]. The existing literature on relaxation of variational problems, including two monographs by Buttazzo [3] and Dacorogna [9], and the review paper by Marcellini [18] containing a considerable list of references, is quite rich. However, the author failed to And a setting similar to that of the paper. For the most recent results on relaxation and related topics see [1,4–8,11,14]. This paper deals with the case where an integrand depends on a scalar function of several variables. At the end of the paper we will make a conjecture on generalization of the main relaxation result of the paper to the case of an integrand depending on a vector function of several variables. We also make a conjecture on generalization of

Morse lemmas for multi-dimensional variational problems

Morse lemmas for multi-dimensional variational problems

ON THE QUESTION OF EXTENSION OF MULTIDIMENSIONAL VARIATIONAL PROBLEMS

The extension of multidimensional variational problems is investigated. In particular, a multidimensional analogue of the Bogolyubov theorem on extension of a simple problem of the calculus of variations is proved. The problem of the closure of the class of functions of several variables with derivatives in a given set is considered. This problem is closely connected with the extension of variational problems. Bibliography: 16 titles.

Konvexe Bereiche kleinster Oberfläche bei gegebener Dicke

A till now unconfirmed conjecture is proved analytically: The balls are in \mathbb E^n (n \geq 3) the single convex domains of given thickness with smallest surface area. The proof is effected by means of a generalized duality theory of restricted multidimensional variational problems in parametric form.

Lower semicontinuous extension of multidimensional variational problems

Lower semicontinuous extension of multidimensional variational problems

Coleman's principle for the determination of the stationary points of invariant functions

One finds broad conditions on a set X, on a collection G of operators acting on X, and on a functional f, defined on X and invariant relative to G, under which the following fact holds: a point which is stationary for f on Xo, the set of all fixed (relative to G) points from X, is stationary also for f on the entire X. One gives applications of the theorems established in the paper to various multidimensional variational problems, among others, to the Skyrme model.

Discrete-sample curve fitting using chebyshev polynomials and the approximate determination of optimal trajectories via dynamic programming

Some useful properties of the Chebyshev polynomials are derived. By virtue of their discrete orthogonality, a truncated Chebyshev polynomials series is used to approximate a function whose discrete samples are the only available data. If minimization of the sum of the discrete squared error is used as the criterion, subject to some constraints on initial conditions and/or terminal conditions, the coefficients of the polynomials are easy to obtain. The simplicity of computing the coefficients of the polynomials from the discrete values of the function to be approximated is utilized to the approximate determination of optimal trajectories via dynamic programming using the technique of polynomial approximation. This allows use of the functional equation approach to solve multi-dimensional variational problems.