Degenerate Matrix Research Articles

Exosystem‐observer approach to H 2 control with output regulation constraint

This study proposes an exosystem-observer approach to an H 2 output feedback problem with an output regulation constraint. The authors analyse the error covariance of a recently proposed exosystem observer and newly derive the degenerate Riccati equation and linear matrix inequalities for designing the optimal and suboptimal exosystem observer gains. Furthermore, they show that the H 2 output feedback controller based on the exosystem observer is equivalently implemented in a Youla-parameter-based structure, where the Youla parameter is realised as an observer for the exosystem state. By utilising the Youla-parameter-based structure, they provide output feedback controller design methods for achieving the optimal and suboptimal H 2 performances.

On the Optimal Control Problem to Solutions of One Granberg Model

The active development of methods for solving inhomogeneous systems of differential equations with a degenerate matrix with a derivative is primarily associated with a wide range of applied problems. Optimal control to solutions of these problems is also an important area of research. The article considers the problem on optimal control to solutions of the non-stationary Granberg model. The main methods of the study are methods of the theory of degenerate (semi) groups and optimal control for Sobolev type equations. The given example of solving the problem from the monograph written by A. G. Granberg illustrates the advantages of the applied methods for solving. Namely, the methods do not require the introduction of assumptions that were applied earlier and do not correspond to real situations when solving such problems. Also, as an example, we give an exact solution to the optimal control problem in which the planned values of economic indicators are taken in the form of a second-order polynomial with a control action in the form of a third-order polynomial. In addition, we propose an algorithm for numerically solving the optimal control problem under consideration.

Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces

We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the generator of the Feller process, we are able to define a notion of absolutely continuous trajectories of measures in terms of some topology on this core. Also, we define a Hamiltonian in terms of the linear generator and a Lagrangian as its Legendre transform. We prove the large deviation principle and show that the rate function can be decomposed as a rate function for the initial time and an integral over the Lagrangian, finite only for absolutely continuous trajectories of measures. We apply this result for diffusion and Lévy processes on $\mathbb{R}^{d}$, for pure jump processes with bounded jump kernel on arbitrary locally compact spaces and for discrete interacting particle systems. For diffusion processes, the theorem partly extends the Dawson and Gärtner theorem for noninteracting copies in the sense that it only holds for time-homogeneous processes, but on the other hand it holds for processes with degenerate diffusion matrix.

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.

How a decreased fibrillar interconnectivity influences stiffness and swelling properties during early cartilage degeneration.

The functional coupling between the fibrillar network and the high-swelling proteoglycans largely determines the mechanical properties of the articular cartilage matrix. The objective of this new study was to show specifically how changes in fibrillar interconnectivity arising from early cartilage degeneration influence transverse stiffness and swelling properties at the tissue level. Radial zone transverse layers of cartilage matrix were obtained from intact and mildly degenerate bovine patellae. Each layer was then subdivided to assess tensile stiffness, free-swelling response, glycosaminoglycan (GAG) content, and micro- and ultra-structural features. The tensile modulus was significantly lower and the degree of swelling significantly higher for the degenerate matrix compared to the intact. Scanning electron microscopy revealed a homogeneous response to transverse strain in the intact cartilage, whereas large non-fibrillar spaces between fibril aggregates were visible in the degenerate matrix. Although there were no significant differences in GAG content it did correlate significantly with stiffness and swelling in the intact samples but not in the degenerate. The lower degree of fibril network interconnectivity in the degenerate matrix led to both a decreased transverse stiffness and reduced resistance to osmotic swelling. This network 'de-structuring' also resulted in a reduced functional interaction between the fibrillar network and the proteoglycans. The study provides new insights into the role of the fibrillar network and how changes in the network arising from the degenerative cascade will influence tissue level behaviour.

Fokker–Planck–Kolmogorov equations with a partially degenerate diffusion matrix

The Fokker–Planck–Kolmogorov equations with a degenerate or partially degenerate diffusion matrix are considered. The distance between probability solutions of these equations with different drift coefficients and different initial conditions is estimated. Sufficient conditions for the existence and uniqueness of probability solutions to nonlinear Fokker–Planck–Kolmogorov equations with a partially degenerate diffusion matrix are established.

On the Construction of Asymptotics of the Solution of Multipoint Boundary-Value Problem for a Linear Degenerate Singularly Perturbed System of Differential Equations

We study the possibility of construction of the asymptotic solution of multipoint boundary-value problem for a linear singularly perturbed system of differential equations with identically degenerate matrix of derivatives in the case of multiple spectrum of the main operator. To this end, we use the results of asymptotic analysis for the general solution of a linear degenerate singularly perturbed system of differential equations. We establish the existence and uniqueness conditions for the solution of the analyzed boundary-value problem.

Asymptotics of the General Solution of Linear Singularly Perturbed Systems of Higher-Order Differential Equations with Degeneration in the Case of Multiple Spectrum of the Limiting Matrix Pencil

We construct an asymptotic expansion for the fundamental system of solutions of a linear singularly perturbed system of m-order differential equations with degenerate principal matrix at the higher-order derivatives. The case where the corresponding characteristic polynomial has a multiple spectrum is investigated. It is proved that, in this case, the asymptotic expansions are constructed in the fractional powers of the small parameter. The recurrence formulas for the coefficients of these expansions are obtained.

Viscosity methods giving uniqueness for martingale problems

Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes. We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $ D$ to be nonsmooth and the direction of reflection to be degenerate. Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix.

Simultaneous diagonalization of three real symmetric matrices

We formulate and prove necessary and sufficient conditions of simultaneous diagonalization of three real symmetric matrices of regular pencil. The conditions are algebraic and consist, in particular, of two spectral requirements and one matrix equality. For the degenerate matrix pencil we suggest an approach that allows reducing of the analysis to a regular pencil. With the use of obtained theorems we investigate a decomposition of linear gyroscopic system into subsystems of an order not higher than two and the stability of trivial solution to a system.

On qualitative properties of differential-algebraic equations

Linear systems of ordinary differential equations with identically degenerate coefficient matrix before the derivative of the unknown vector function are considered. The structure of general solutions and the notion of singular point of such systems are discussed. From the comparison of the properties of the “perturbed” and original problems, a sufficient criterion for the Lyapunov asymptotic stability of the zero solution is obtained.

Узагальнений аналіз матричних представлень асоціативних гіперкомплексних числових систем, що використовуються в енергетиці

З найбільш загальних позицій розглянуто питання представлення комплексних чисел, кватер ніонів, квадріплексних (бікомплексних) чисел і бікватерніонів комплексними матрицями другого порядку. Для побудови матричних базисів зображень, що розглядаються, з множини зазначених матриць виділено підмножину елементів, квадрат яких дорівнює від’ємній матричній одиниці. Показано, що внутрішня структура вироджених уявних матричних одиниць дозволяє конструювати числові системи, не вдаючись до аксіоматичного визначення алгебраїчної операції множення. Сформульовано ряд тверджень, які дають змогу спростити вирішення порушених у статті питань. Бібліогр.: 14 найм.

Differential-geometric aspects of a nonholonomic Dirac mechanics: Lessons of a model quadratic in velocities

Faddeev and Vershik proposed the Hamiltonian and Lagrangian formulations of constrained mechanical systems that are invariant from the differential geometry standpoint. In both formulations, the description is based on a nondegenerate symplectic 2-form defined on a cotangent bundle T*Q (in the Hamiltonian formulation) or on a tangent bundle TQ (in the Lagrangian formulation), and constraints are sets of functions in involution on these manifolds. We demonstrate that this technique does not allow “invariantization” of the Dirac procedure of constraint “proliferation.” We show this in an example of a typical quantum field model in which the original Lagrange function is a quadratic form in velocities with a degenerate coefficient matrix. We postulate that the initial phase space is a manifold where all arguments of the action functional including the Lagrange multipliers are defined. The Lagrange multipliers can then be naturally interpreted physically as velocities (in the Hamiltonian formulation) or momenta (in the Lagrangian formulation) related to “nonphysical” degrees of freedom. A quasisymplectic 2-form invariantly defined on such a manifold is degenerate. We propose new differential-geometric structures that allow formulating the Dirac procedure invariantly.

On the solvability and numerical methods for solution of linear integro-algebraic equations

We study solvability conditions for a system of Volterra equations with some identically degenerate or rectangular matrix at the main term. Connection is discussed of the solvability conditions and applicability of numerical methods for solving these systems. In particular, the conditions of the convergence of the least squares method with the error functional defined in Sobolev spaces are presented.

On the structure of the general solution and conditions of solvability of the Cauchy problem for degenerate linear systems of higher-order differential equations

For a system of linear differential equations of order p with identically degenerate coefficient matrix of higher derivatives, we establish conditions under which this system has the general solution of the Cauchy type. The structure of this solution is determined. Conditions for the existence and uniqueness of solution of the corresponding initial-value problem are also obtained.

Eigendecomposition of the Discrete Double-Curl Operator with Application to Fast Eigensolver for Three-Dimensional Photonic Crystals

This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three-dimensional photonic crystals with face-centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP, and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a precon...

Testing for Common Conditionally Heteroskedastic Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that Hansen's (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a fifty-fifty mixture of ... and ..., where H is the number of moment conditions, as opposed to a .... With more than two assets, this distribution lies between the ... and ... (p denotes the number of parameters). These results show that ignoring the lack of first-order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets. A Monte Carlo study illustrates these findings. (ProQuest: ... denotes formulae/symbols omitted.)

Testing for Common GARCH Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that the Hansen’s (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain empirical process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a half-half mixture of x_(H-1)^2and x_H^2, where H is the number of moment conditions, as opposed to a x_(H-1)^2. With more than two assets, this distribution lies between the x_(H-p)^2 and x_H^2 (p, the number of parameters). These results show that ignoring the lack of first order identification of the moment condition model leads to oversized tests with possibly increasing over-rejection rate with the number of assets. A Monte Carlo study illustrates these findings.

Matrix representation of the double-curl operator for simulating three dimensional photonic crystals

Three dimensional photonic crystals can be modeled by the Maxwell equations as a generalized eigenvalue problem (GEVP). Simulations based on the numerical solutions of the GEVP are used to reveal physical properties and boost innovative applications of photonic crystals. However, to solve these GEVP remains a computational challenge in both timing and accuracy. The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice is one of the challenging eigenvalue problems. From a viewpoint of matrix computation, we demonstrate how such obstacles can be overcome. Our main contribution is an explicit matrix representation of the double-curl operator associated with the photonic crystal with FCC lattice. This particular matrix represents the degenerate coefficient matrix of the discrete GEVP obtained by Yee’s scheme. The explicit matrix leads to an eigendecomposition of the degenerate coefficient matrix and then a fast eigenvalue solver. Promising numerical results in terms of timing and accuracy are reported for solving the discrete GEVP arising in three dimensional photonic crystals with various geometric parameters.

Mimicking the respiratory burst myeloperoxidase system of polymorphonuclear neutrophil granulocytes to aid the wound healing behaviors of articular cartilage

Purpose: Articular cartilage exhibits a limited ability to remove damaged tissue from its surfaces, a step integral for efficient mammalian wound healing that is typically enabled by polymorphonuclear neutrophil granulocytes in other tissue types. This limited ability constrains articular cartilage wound healing behaviors, contributing to lesion progression and the disease burden of osteoarthritis. The respiratory burst myeloperoxidase system of polymorphonuclear neutrophil granulocytes produces low stability protonating agents involved in exothermic tissue homeostasis and repair mechanisms through disproportionation redox reactions. Because these protonating agents facilitate the removal of damaged interstitial matrices through biopolymer disaggregation in order to assist differentiated tissue assembly activities, this study evaluates the capability of alternating current redox magnetohydrodynamic technology to mimic this respiratory burst to aid the wound healing behaviors of articular cartilage by creating a healthy lesion site devoid of damaged tissue. Methods: An alternating current redox magnetohydrodynamic device fashioned as a physiochemical scalpel was deployed in 0.9% sodium chloride solutions at 300 mOsm/L and at 20oC. Figure 1 demonstrates the device producing solution changes that are delivered to tissue surfaces as an engineered irrigant during endoscopic procedures designed to stabilize articular cartilage lesions. To determine if the engineered irrigant resembles the protic solvent generated by azurophilic granules, the resultant irrigant temperature and protonation potential were measured as a function of power input during device activation controlled for 5 second steady-state treatment conditions. Irrigant protonation potential was determined by measuring solution electrochemical potential relative to [H+] as a function of differential proton sequestration in the irrigant during device activation. Results: Alternating current redox magnetohydrodynamics in saline solutions is represented in Figure 2. As depicted in Figure 3, the protonation potential increased with direct correlation to power delivery (p < 0.02; R2 = 0.311) and commensurate with a minimal change in irrigant temperature (∼0-5o C) above the baseline 20o C, reflecting features characteristic of the protic solvent generated by the azurophilic degranulation of polymorphonuclear neutrophil granulocytes during the early phases of wound healing. Conclusions: Because resection precision that eliminates both volumetric and functional over-resection is required before surgical lesion stabilization can be an effective aid to articular cartilage wound healing behaviors, polymorphonuclear neutrophil granulocyte function is a valuable therapeutic design resource. Protonation coupled conformational dynamics is an energy transduction processes that achieves nanometer resection precision through a guest chemical denaturization process below the isoelectric point of exposed damaged interstitial tissue matrices. Because of high proton motilities in water solutions, stoichiometric protonation is a very rapid charge redistribution process that leads to biopolymer disaggregation through molecular cleavage planes accessible due to normal tissue surface barrier losses and degenerate matrix properties characteristic of damage tissue sites. Figure 4 illustrates a representative integrated cell viability stain section image that demonstrates removing the bioburden of damaged tissue without iatrogenic over-resection in a human explant model of osteoarthritis by adapting alternating current redox magnetohydrodynamic technology for tissue rescue surgical procedures.