Solvability Analysis Research Articles

Optimal Control of Nonlinear Transport-Flow Mixed Variational Problems

Optimal control of nonlinear transport-flow mixed variational problems are studied qualitatively, and the solvability analysis of the transport and flow mixed state systems is performed on the basis of primal and dual evolution duality principles. Corresponding primal and dual mixed optimality conditions are established by the application of some fundamental perturbation conjugate duality results recently proposed [8]. Further, for computational purposes, two- and three-field proximation penalty-duality algorithms in the resolution of the mixed optimality conditions are finally presented and discussed.

Solvability analysis of a nonlocal boundary value problem by applying the contraction mapping principle

The existence and uniqueness of a classical solution to the nonlocal boundary value problem for Poisson’s operator on a two-dimensional rectangular domain is proved in detail by applying the contraction mapping principle.

Optimal Control of Evolution Mixed Variational Inclusions

Optimal control problems of primal and dual evolution mixed variational inclusions, in reflexive Banach spaces, are studied. The solvability analysis of the mixed state systems is established via duality principles. The optimality analysis is performed in terms of perturbation conjugate duality methods, and proximation penalty-duality algorithms to mixed optimality conditions are further presented. Applications to nonlinear diffusion constrained problems as well as quasistatic elastoviscoplastic bilateral contact problems exemplify the theory.

Application of perturbation theory to the solvability analysis of differential algebraic equations

Linear systems of ordinary differential equations with an identically singular or rectangular matrix multiplying the derivative of the desired vector function are considered. The structure of general solutions is discussed. A special case of perturbed systems is studied by applying the Vishik-Lyusternik method.

A priori and a posteriori error analyses of augmented twofold saddle point formulations for nonlinear elasticity problems

In this paper we introduce and analyze new augmented mixed finite element methods allowing much more flexibility in the choice of the corresponding finite dimensional subspaces for a class of nonlinear elasticity problems arising in hyperelasticity. The starting mixed method is based on the incorporation of the strain tensor as an auxiliary unknown, which, together with the usual stress-displacement-rotation approach employed in linear elasticity, yields a nonlinear twofold saddle point operator equation as the resulting weak formulation. We first extend known results on the well-posedness of the associated Galerkin scheme with PEERS of order k=0 to the case k⩾1. Then the augmented schemes are obtained by adding consistent Galerkin-type terms arising first from the constitutive equation, and then from the equilibrium equation and the relations defining the rotation in terms of the displacement and the strain tensor as independent unknown, all of them multiplied by suitably chosen stabilization parameters. We apply classical results on the solvability analysis of nonlinear saddle point and strongly monotone operator equations to prove that the corresponding continuous and discrete augmented schemes are well-posed. More precisely, and as the main advantages of our present approach, we show that the well-posedness of a partially augmented Galerkin scheme is ensured by any finite element subspace for the strain tensor together with the PEERS space of order k⩾0 for the remaining unknowns, whereas any finite element subspace of the whole continuous space will do in the case of a fully augmented scheme. Then, we derive reliable and efficient residual-based a posteriori error estimators for all the schemes. Finally, we provide several numerical results illustrating the good performance of the mixed finite element methods, confirming the theoretical properties of the estimators, and showing the behaviour of the associated adaptive algorithms.

Passivity-Based Control Scheme for the Modular Multilevel Cascaded Converter (MMCC) as the HVDC-PLUS for Smart Grid Applications

This paper presents the passivity-based controller design methodologies of the 3-phase modular multilevel cascaded converter (MMCC) as HVDC-PLUS for smart grid applications. The mathematical model of the MMCC is derived in the terms of energy dissipation mechanism among the converter cells. The solvability and passivity analysis are presented based on the state-space averaged (SSA) models. The energy shaping (ES) problem is formulated by properly selection of the energy function. And the dissipative characteristics of the MMCC-based HVDC-PLUS is revealed. The interconnection and damping assignment (IDA) technique is proposed to reshape the trajectory of the control objective to guarantee the global convergence and accurate reference tracking. The carrier phase-shifted PWM scheme is adopted to synthesize the control signals for the individual half-bridge converters. The various simulation results under reactive and active power transmission scenarios are presented, which confirms the effectiveness of the proposed passivity-based control methodologies

Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints

The problem of calculating joint reaction forces in rigid body mechanisms with redundant constraints, both geometric and nonholonomic, is discussed. When constraint equations are dependent, some of the constraint reactions are unsolvable, i.e., cannot be uniquely determined using a rigid body model, whereas some others may be solvable. In this paper, analytic conditions, which must be fulfilled to obtain unique values of selected reaction forces in the presence of dependent nonholonomic constraints, are presented and proven. The concept of direct sum, known from linear algebra, is exploited. These purely mathematical conditions are followed by numerical methods that enable detection of constraints with uniquely solvable reactions. Similar conditions and methods were proposed earlier for holonomic systems. In this contribution, they are generalized to the case of linear nonholonomic constraints. An example of constraint reactions solvability analysis, for a mechanism subjected to redundant nonholonomic constraints, is presented.

Solvability of Variational Inequalities on Hilbert Lattices

This paper provides a systematic solvability analysis for (generalized) variational inequalities on separable Hilbert lattices. By contrast to a large part of the existing literature, our approach is lattice-theoretic, and is not based on topological fixed point theory. This allows us to establish the solvability of certain types of (generalized) variational inequalities without requiring the involved (set-valued) maps be hemicontinuous or monotonic. Some of our results generalize those obtained in the context of nonlinear complementarity problems in earlier work, and appear to have scope for applications. This is illustrated by means of several applications to fixed point theory, optimization, and game theory.

Integrable variational equations of non-integrable systems

Paper is devoted to the solvability analysis of variational equations obtained by linearization of the Euler-Poisson equations for the symmetric rigid body with a fixed point on the equatorial plain. In this case Euler-Poisson equations have two pendulum like particular solutions. Symmetric heavy top is integrable only in four famous cases. In this paper is shown that a family of cases can be distinguished such that Euler-Poisson equations are not integrable but variational equations along particular solutions are solvable. The connection of this result with analysis made in XIX century by R. Liouville is also discussed.

On The Anisotropic Wiener-hopf Operator, Connected With Helmholtze-sohrodinger Equation

In this article, solvability of one the anisotropic Helmholtz-Shrodinger equation with the boundary conditions of the first and second type is investigated in the upper and lower half –space, (x5>0, x5<0), in 5 dimensions. Solvability of these boundary problems reduces to solvability of Rieman- Hilbert boundary problem, in general necessary and sufficient conditions for the correctness of the problem in the Sobolev space are presented as well as explicit formulas for a factorization of the Fourier symbol matrix of the one-medium problem. The solvability analysis is based on the factorization problem of some matrix-function.

Evolution Mixed Variational Inclusions with Optimal Control

Evolution mixed maximal monotone variational inclusions with optimal control, in reflexive Banach spaces, are analized. Solvability analysis is performed on the basis of composition duality principles. Applications to nonlinear diffusion constrained problems, as well as to quasistatic elastoviscoplastic contact problems exemplify the theory.

On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction in joints

The uniqueness of simulated motion of an overconstrained rigid body mechanism with joint friction is studied. The investigated issue originates in the problem of joint reactions solvability. It is known that in case of redundant constraints existence the constraint reaction forces cannot be — in general — uniquely determined. It can be proved, however, that — under certain conditions — selected reactions can be specified uniquely. Analytical and numerical methods for reactions solvability analysis are available. It is shown in this paper that indeterminacy of normal reactions results in indeterminacy of friction forces, and moreover, non-uniqueness of friction forces results in non-uniqueness of simulated motion. A method of finding these joints, for which friction forces are unique, is presented. It is also proved that if only uniquely solvable friction effects are introduced, then simulated motion of the mechanism is unique, otherwise it is not. Finally, examples of dynamic analysis of overconstrained mechanisms with joint friction are presented; unique and non-unique results are obtained.

Moving-boundary approximation for curved streamer ionization fronts: Numerical tests

Recently a moving boundary approximation for the minimal model for negative streamer ionization fronts was extended with effects due to front curvature; this was done through a systematic solvability analysis. A central prediction of this analysis is the existence of a nonvanishing electric field in the streamer interior, whose value is proportional to the front curvature. In this paper we compare this result and other predictions of the solvability analysis with numerical simulations of the minimal model.

On the technique for passing to the limit in nonlinear elliptic equations

We consider the problem of passing to the limit in a sequence of nonlinear elliptic problems. The “limit” equation is known in advance, but it has a nonclassical structure; namely, it contains the p-Laplacian with variable exponent p = p(x). Such equations typically exhibit a special kind of nonuniqueness, known as the Lavrent’ev effect, and this is what makes passing to the limit nontrivial. Equations involving the p(x)-Laplacian occur in many problems of mathematical physics. Some applications are included in the present paper. In particular, we suggest an approach to the solvability analysis of a well-known coupled system in non-Newtonian hydrodynamics (“stationary thermo-rheological viscous flows”) without resorting to any smallness conditions.

Covering mappings and their applications to differential equations unsolved for the derivative

We continue to study the properties of covering mappings of metric spaces and present their applications to differential equations. To extend the applications of covering mappings, we introduce the notion of conditionally covering mapping. We prove that the solvability and the estimates for solutions of equations with conditionally covering mappings are preserved under small Lipschitz perturbations. These assertions are used in the solvability analysis of differential equations unsolved for the derivative.

Moving boundary approximation for curved streamer ionization fronts: Solvability analysis

The minimal density model for negative streamer ionization fronts is investigated. An earlier moving boundary approximation for this model consisted of a "kinetic undercooling" type boundary condition in a Laplacian growth problem of Hele-Shaw type. Here we derive a curvature correction to the moving boundary approximation that resembles surface tension. The calculation is based on solvability analysis with unconventional features, namely, there are three relevant zero modes of the adjoint operator, one of them diverging; furthermore, the inner-outer matching ahead of the front must be performed on a line rather than on an extended region; and the whole calculation can be performed analytically. The analysis reveals a relation between the fields ahead and behind a slowly evolving curved front, the curvature and the generated conductivity. This relation forces us to give up the ideal conductivity approximation, and we suggest to replace it by a charge neutrality approximation. This implies that the electric potential in the streamer interior is no longer constant but solves a Laplace equation; this leads to a Muskat-type problem.

On mixed boundary value problems for pseudoparabolic systems

Solvability analysis of mixed boundary value problems for pseudoparabolic systems in a special scale of weighted Sobolev spaces is presented. The class under consideration contains the linearized Navier-Stokes system. It is proved that, choosing the power weight, one can diminish the number of solvability conditions and in some cases obtain unconditional solvability of the boundary value problems.

Sequential blind extraction of instantaneously mixed sources

This paper presents a general approach to sequential blind extraction of instantaneously mixed sources for several major ill-conditioned cases as well as the regular case of full column rank mixing matrices. Four ill-conditioned cases are considered: The mixing matrix is square but singular; the number of sensors is less than that of sources; the number of sensors is larger than that of sources, but the column rank of the mixing matrix is deficient; and the number of sources is unknown and the column rank of the mixing matrix is deficient. First, a solvability analysis is presented for a general case. A necessary and sufficient condition for extractability is derived. A sequential blind extraction approach is then proposed to extract all theoretically separable sources. Next, a principle and a cost function based on fourth-order cumulants are presented for blind source extraction. By minimizing the cost function under a nonsingularity constraint of the extraction matrix, all theoretically separable sources can be extracted sequentially. Finally, simulation results are presented to demonstrate the validity and performance of the blind source extraction approach.

Structural solvability analysis of dynamic process models

The variable structure of dynamic process models is represented by a directed graph termed as the representation graph for the purpose of solvability analysis in this paper. Structural solvability analysis, the determination of the structural differential index and the structural decomposition of the differential–algebraic equations (DAE) model set can be performed using the representation graph. The characteristic features of the representation graph for both index 1 and high index semi-explicit DAE models are presented. Based on the above a novel index reduction procedure for high index models is proposed. The notions and methods are illustrated on simple process examples.

A computationally implementable criterion for the solvability of boundary value problems on an infinite domain

In this note we develop a simple, computationally implementable criterion to determine the solvability of boundary value problems. This criterion for solvability involves converting the boundary value problem to an initial value problem, forming an associated nonlinear boundary constraint, and computing necessary conditions for convergence using a coupled variational initial value problem. Source term and near-field boundary condition effects are explicitly included. The solvability analysis is directly analogous to sampling convergence of a classical shooting method. Analytical and computational examples are provided which give confidence that this methodology is robust and has potentially wide applicability. Extensions to systems of equations are provided and demonstrated for a linear problem.