Controlling Stochastic Sensitivity by Feedback Regulators in Nonlinear Dynamical Systems with Incomplete Information

The problem of synthesis of stochastic sensitivity for equilibrium modes in nonlinear randomly forced dynamical systems with incomplete information is considered. We construct a feedback regulator that uses noisy data on some system state coordinates. For parameters of the regulator providing assigned stochastic sensitivity, a quadratic matrix equation is derived. Attainability of the assigned stochastic sensitivity is reduced to the solvability of this equation. We suggest a constructive algorithm for solving this quadratic matrix equation. These general theoretical results are used to solve the problem of stabilizing equilibrium modes of nonlinear stochastic oscillators under conditions of incomplete information. Details of our approach are illustrated on the example of a van der Pol oscillator.

About a fixed-point-type transformation to solve quadratic matrix equations using the Krasnoselskij method.

In this paper, we study the simplest quadratic matrix equation: $\mathcal{Q}(X)=X^2+BX+C=0$. We transform this equation into an equivalent fixed-point equation and based on it we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor-corrector iterative scheme with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improves the numerical application of the Newton method when approximating a solution of the quadratic matrix equation.

A numerically exact non-reflecting boundary condition applied to the acoustic Helmholtz equation

When modeling wave propagation, truncation of the computational domain to a manageable size requires non-reflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to infinity. Constant extrapolation in the direction perpendicular to the boundary provides the material properties in the exterior. Domain decomposition can split the enlarged domain into the original one and its exterior. Because the boundary-value problem for the latter is translation-invariant, the boundary Green functions obey a quadratic matrix equation. Selection of the solvent that corresponds to the outgoing waves provides the input for the remaining problem in the interior. The result is a numerically exact non-reflecting boundary condition on one side of the domain. When two non-reflecting sides have a common corner, translation invariance is lost. Treating each side independently in combination with a classic absorbing condition in the other direction restores translation invariance and enables application of the method at the expense of numerical exactness. Solving the quadratic matrix equation with Newton's method turns out to be more costly than solving the Helmholtz equation and may select unwanted incoming waves. A proposed direct method has a much lower cost and selects the correct branch. A test on a 2-D acoustic marine seismic problem with a free surface at the top, a classic second-order Higdon condition at the bottom and numerically exact boundaries at the two lateral sides demonstrates the capability of the method. Numerically exact boundaries on each side, each computed independently with a free-surface or Higdon condition, provide even better results.

Fast verified computation for the solvent of the quadratic matrix equation

Two fast algorithms for numerically computing an interval matrix containing the solvent of the quadratic matrix equation $AX^2 + BX + C = 0$ with square matrices $A$, $B$, $C$ and $X$ are proposed. These algorithms require only cubic complexity, verify the uniqueness of the contained solvent, and do not involve iterative process. Let $\ap{X}$ be a numerical approximation to the solvent. The first and second algorithms are applicable when $A$ and $A\ap{X}+B$ are nonsingular and numerically computed eigenvector matrices of $\ap{X}^T$ and $\ap{X} + \inv{A}B$, and $\ap{X}^T$ and $\inv{(A\ap{X}+B)}A$ are not ill-conditioned, respectively. The first algorithm moreover verifies the dominance and minimality of the contained solvent. Numerical results show efficiency of the algorithms.

Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.

On the commuting solutions to the Yang-Baxter-like matrix equation for identity matrix minus special rank-two matrices

Let A=I-PQT, where P and Q are two n x 2 complex matrices of full column rank such that det(QTP)=0. We find all the commuting solutions of the quadratic matrix equation AXA = XAX.

On the Yang-Baxter-like matrix equation for rank-two matrices

Let A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that QTP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.