Necessary Condition For Solvability Research Articles

Fault Detection and Isolation for Linear Structured Systems

This paper deals with the fault detection and isolation (FDI) problem for linear structured systems in which the system matrices are given by zero/nonzero/arbitrary pattern matrices. In this paper, we follow a geometric approach to verify solvability of the FDI problem for such systems. To do so, we first develop a necessary and sufficient condition under which the FDI problem for a given particular linear time-invariant system is solvable. Next, we establish a necessary condition for solvability of the FDI problem for linear structured systems. In addition, we develop a sufficient algebraic condition for solvability of the FDI problem in terms of a rank test on an associated pattern matrix. To illustrate that this condition is not necessary, we provide a counterexample in which the FDI problem is solvable while the condition is not satisfied. Finally, we develop a graph-theoretic condition for the full rank property of a given pattern matrix, which leads to a graph-theoretic condition for solvability of the FDI problem.

Generalization of Plane Stress and Plane Strain States to Elastic Plates of Finite Thickness

This paper presents a novel method to establish a general solution for an isotropic homogeneous elastic plate of finite thickness. Under the assumption of vanishing out-of-plane shear stresses, a necessary condition of solvability of elastic problems is obtained. Moreover, a general solution dependent on the thickness-wise coordinate is derived, where the unknown function is still governed by a two-dimensional biharmonic equation. In terms of the two-dimensional Airy stress function or the classical solution in plane strain state, exact elastic stresses in an elastic plate of finite thickness are given. The solution for plane strain state can be a special case of the present, whereas that classic plane stress state is reduced by letting thickness approach zero. A comparison of the results between the present paper and plane stress state is made.

Hermitian solutions to the system of operator equations T_iX=U_i.

In this article we consider the system of operator equations T_iX=U_i for i=1,2,...,n and give necessary and suffcient conditions for the existence of common Hermitian solutions to this system of operator equations for arbitrary operators without the closedness condition. Also we study the Moore-penrose inverse of a ncross 1 block operator matrix and. then give the general form of common Hermitian solutions to this system of equations. Cosequently, we give the necessary and sffcient conditions for the existence of common Hermitian solutions to the system of operator equati and also present the necessary conditions for solvability of the equation sum_{i=1}{n}T_iX_i=U

On Necessary Conditions for the Solvability of One Class of Elliptic Systems in a Half-Space

We consider boundary value problems in a half-space for a class of elliptic systems. Assuming that the boundary value problems satisfy the Lopatinskii˘ condition, we give necessary conditions for the unique solvability in Sobolev spaces.

Necessary condition for solvability of the control problem of an asynchronous spectrum of linear almost periodic systems with trivial averaging of the coefficient matrix

We consider a linear control system with an almost periodic matrix of coefficients. The control has a form of feedback and is linear in phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i.e. the smallest additive group of real numbers, including all Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix.The following problem is formulated: choose such a control from an admissible set so that the closed system has almost periodic solutions, the frequency spectrum (a set of Fourier exponents) of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The problem is called the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies.The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix.

Unilateral contact problems with a friction

The boundary contact problem for a micropolar homogeneous elastic hemitropic medium with a friction is investigated. Here, on a part of the elastic medium surface with a friction, instead of a normal component of force stress there is prescribed the normal component of the displacement vector. We give their mathematical formulation of the Problem in the form of spatial variational inequalities. We consider two cases, the so-called coercive case (when elastic medium is fixed along some part of the boundary) and semi-coercive case (the boundary is not fixed). Based on our variational inequality approach, we prove the existence and uniqueness theorems and show that solutions continuously depend on the data of the original problem. In the semi-coercive case, the necessary condition of solvability of the corresponding contact problem is written out explicitly. This condition under certain restrictions is sufficient, as well.

Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane

This note is mainly devoted to the solvability of Monge-Ampere equation det ( D2u ) = 1, in a C2 bounded strictly convex domain Ω ⊂ R2, subject to a contact angle boundary condition. A necessary condition for the solvability of this problem, involving the maximal value of the curvature k (s) of ∂Ω and the contact angle, was derived by X.-N. Ma in [10], making use of a maximum principle for an appropriate P-function (see, also, J. Urbas [20], for a different proof). Our main goal here is to prove a complementary result. More precisely, we will derive a new necessary condition of solvability, involving the minimal value of the curvature k (s) of ∂Ω and the contact angle. The main ingredients of our proof are the derivation of a minimum principle for the P-function employed by X.-N. Ma in his proof, respectively the use of some computations in normal coordinates with respect to the boundary ∂Ω. Finally, a similar minimum principle will be employed to derive some isoperimetric estimates for the classical convex solution of Monge-Ampere equation, subject to the homogeneous Dirichlet boundary condition.

Optimal control problem for stationary quasi-optic equations

In this paper, an optimal control problem was taken up for a stationary equation of quasi optic. For the stationary equation of quasi optic, at first judgment relating to the existence and uniqueness of a boundary value problem was given. By using this judgment, the existence and uniqueness of the optimal control problem solutions were proved. Then we state a necessary condition to an optimal solution. We proved differentiability of a functional and obtained a formula for its gradient. By using this formula, the necessary condition for solvability of the problem is stated as the variational principle.

Extremal holomorphic maps and the symmetrized bidisc

We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $\Gamma$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence $\mathcal{C}_\nu, \nu \geq 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show that $\mathcal{C}_{n-3}$ is insufficient for the solvability of an $n$-point problem for $n\geq 3$. We propose the conjecture that condition $\mathcal{C}_{n-2}$ is necessary and sufficient for the solvability of an $n$-point interpolation problem for $\Gamma$ and we explore the implications of this conjecture. We introduce a classification of rational $\Gamma$-inner functions, that is, analytic functions from the disc into $\Gamma$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $\Gamma$. The classes are related to $n$-extremality and the conditions $\mathcal{C}_\nu$; we prove numerous strict inclusions between the classes.

Boundary singularities of solutions to elliptic viscous Hamilton–Jacobi equations

We study the boundary value problem with measures for (E1) −Δu+g(|∇u|)=0 in a bounded domain Ω in RN, satisfying (E2) u=μ on ∂Ω and prove that if g∈L1(1,∞;t−(2N+1)/Ndt) is nondecreasing (E1)–(E2) can be solved with any positive bounded measure. When g(r)⩾rq with q>1 we prove that any positive function satisfying (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When g(r)=rq with 1<q<qc=N+1N we prove the existence of a positive solution with a general outer regular Borel measure ν≢∞ as boundary trace and characterize the boundary isolated singularities of positive solutions. When g(r)=rq with qc⩽q<2 we prove that a necessary condition for solvability is that μ must be absolutely continuous with respect to the Bessel capacity C2−qq,q′. We also characterize boundary removable sets for moderate and sigma-moderate solutions.

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur–Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna–Pick interpolation problem in the Schur class on the unit polydisk.

𝐿²-well-posedness of 3d div-curl boundary value problems

Criteria for the existence and uniqueness of weak solutions of div-curl boundary-value problems on bounded regions in space with C 2 C^2 -boundaries are developed. The boundary conditions are either given normal components of the field or else given tangential components of the field. Under natural integrability assumptions on the data, finite-energy ( L 2 L^2 ) solutions exist if and only if certain compatibility conditions hold on the data. When compatibility holds, the dimension of the solution space of the boundary-value problem depends on the differential topology of the region. The problem is well-posed with a unique solution in L 2 ( Ω ; R 3 ) L^2(\Omega ;\mathbb {R}^3) provided, in addition, certain line or surface integrals of the field are prescribed. Such extra integrals are described. These results depend on certain weighted orthogonal decompositions of L 2 L^2 vector fields which generalize the Hodge-Weyl decomposition. They involve special scalar and vector potentials. The choices described here enable a decoupling of the equations and a weak interpretation of the boundary conditions. The existence of solutions for the equations for the potentials is obtained from variational principles. In each case, necessary conditions for solvability are described and then these conditions are shown to also be sufficient. Finally L 2 L^2 -estimates of the solutions in terms of the data are obtained. The equations and boundary conditions treated here arise in the analysis of Maxwell’s equations and in fluid mechanical problems.

ON RIEMANN BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ON THE REAL AXIS

In this article, Riemann boundary value problem with different factors for polyanalytic functions on the real axis is studied. The expression of solution and sufficient and necessary condition for solvability of the non-homogeneous Riemann boundary value problem are obtained.

Towards a Unified Approach to Nonexistence of Solutions for a Class of Differential Inequalities

In this paper we deal with an approach for obtaining a priori estimates for solutions of nonlinear partial differential equations and inequalities. The goal is to apply these estimates to find necessary conditions for solvability. This approach, which is simply based of test functions, covers a wide class of nonlinear problems for which we address the question of nonexistence of solutions. In recent years, nonexistence theorems for equations and systems of partial differential equations, have received considerable attention. There is a rather extensive bibliography on this subject. For quasilinear parabolic problems we refer the interested reader to [35],[41], and for more recent results to [20], [7],

Necessary conditions for solvability of a system of linear equations over a ring

Necessary conditions for solvability of systems of linear equations over associative rings are considered. In some cases, these conditions are also sufficient for solvability of systems.

Necessary conditions for solvability of a system of linear equations over a ring

Necessary conditions for solvability of systems of linear equations over associative rings are considered. In some cases, these conditions are also sufficient for solvability of systems.

Linear Forward—Backward Stochastic Differential Equations

The problem of finding adapted solutions to systems of coupled linear forward—backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs.

A necessary condition of solvability for the capillarity boundary of Monge-Ampere equations in two dimensions

In this paper we consider a class of Monge-Ampere equations with a prescribed contact angle boundary value problem on a bounded strictly convex domain in two dimensions. The purpose is to give a sharp necessary condition of solvability for the above mentioned equations. This is achieved by using the maximum principle and introducing a curvilinear coordinate system for Monge-Ampere equations in two dimensions. An interesting feature of our necessary condition is the need for a certain strong restriction between the curvature of the boundary of domain and the boundary condition, which does not appear in the Dirichlet and Neumann boundary values.

A priori bounds and necessary conditions for solvability of prescribed curvature equations

We prove an estimate for the magnitude of solutions of the prescribed higher order mean curvature equations and examine the necessity of our conditions. Our results include well known sharp estimates for the mean and Gauss curvature and our previous estimate for scalar curvature as special cases.

The Hamiltonian system and the spline function

This paper examines the inherent connection between the generalized Hamiltonian system and the spline function. By considering the generalized canonical multipoint boundary value (GCMBV) problem, a multipoint complementary variational principle for generalized Hamiltonian system is provided, and the sufficient and necessary condition for solvability of the GCMBV problem is obtained. With the aid of Hamiltonian equations, a new class of vector spline functions with good variational and approximation properties called H-spline are defined, it includes the general vector spline as a special one. Finally, the H-spline interpolation problems are also discussed.