On a solution of the Hamilton-Jacobi equation arising in molecular biology

We consider the Cauchy problem for the Hamilton-Jacobi equation with phase constraints that arises in molecular biology. We introduce a continuous generalized solution and propose a method of constructing in with the cost function for the auxiliary optimal control problem.     

On the solution of the fuzzy Sylvester matrix equation

In this paper, we consider the fuzzy Sylvester matrix equation AX+XB=C,AX+XB=C, where A ? \mathbbR^{n ×n}A\in {\mathbb{R}}^{n \times n} and B ? \mathbbR^{m ×m}B\in {\mathbb{R}}^{m \times m} are crisp M-matrices, C is an n×mn\times m fuzzy matrix and X is unknown. We first transform this system to an (mn)×(mn)(mn)\times (mn) fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.     

On the solution of a matrix equation in the theory of Luenberger observers

An explicit solution is presented for a matrix equation involved in the design of a Luenberger observer to realize an acceptable approximation to a desired state-variable feedback lawu = u_{0} + Kxfor the systemdx/dt = Ax + Bu, y = Dx.     

Iterations for solving a rational Riccati equation arising in stochastic control

We consider different iterative methods for computing a Hermitian or maximal Hermitian solution of two types rational Riccati equations arising in stochastic control. The classical Newton procedure and its modification applied to equations are very expensive. New less expensive iterations for these equations are introduced and some convergence properties of the new iterations are proved.     

On the solution of a nonlinear matrix equation for MIMO symmetric realizations

A result concerning a particular solution of a nonlinear solution matrix equation reported in a previous paper [2] and interesting for studying SISO systems, is extended to MIMO symmetric realizations. For such a class of systems other important SISO results can be generalized.     

On the numerical solution of a certain class of non-linear cauchy problems arising from integral equation theory

Summary A numerical algorithm for solving a system of non-linear partial differential integral equations is presented. These equations result from applying the method of invariant imbedding to the solution of a certain class ofFredholm integral equations of the second kind [2]. The algorithm is compared with a more standard method of solution and some numerical results presented.     

On the global behavior of a high-order rational difference equation

In this paper, we consider the (k+1)-order rational difference equation

     

On the eigenvalue-eigenvector method for solution of the stationary discrete matrix Riccati equation

The purpose of this correspondence is to point out that certain numerical problems encountered in the solution of the stationary discrete matrix Riccati equation by the eigenvalue-eigenvector method of Vanghan [1] can be avoided by a simple reformulation.     

The numerical solution of the matrix Riccati differential equation

A numerically stable and fast computational method is given for the solution of the matrix Ricatti differential equation with finite terminal time.     

A square root of a matrix approach to obtain the solution to a steady-state matrix Riccati equation

By utilizing the square root of a matrix approach, a method of generating the solution to a steady-state matrix Riccati type equation (under certain restrictions) is presented. This approach not only yields a closed form expression for the Riccati solution, but also converts the original Riccati equation into other equations which may have numerical or computational advantages. An example is worked out for a second-order case.     

On the discrete Lyapunov matrix equation

Some bounds for the arithmetic and the geometric means of the characteristic roots of the positive semidefinite solution to the discrete Lyapunov matrix equation are derived.     

On the Lyapunov matrix differential equation

A lower bound for the determinant of the solution to the Lyapunov matrix differential equation is derived. It is shown that this bound is obtained as a solution to a simple scalar differential equation. In the limiting case where the solution to the Lyapunov differential equation becomes stationary, the result reduces to one of the existing bounds for the algebraic equation.     

Computational experience with the solution of the matrix Lyapunov equation

This correspondence presents a comparative study of three methods for the numerical solution of the matrix Lyapunov equation. The test case is a 24th-order system with highly underdamped eigenvalues and a rather high degree of stiffness. The conclusions favor a method by Bartels and Stewart based on a reduction to Schur form of theAmatrix.     

An iterative algorithm for a least squares solution of a matrix equation

In this article, we propose an iterative algorithm to compute the minimum norm least-squares solution of AXB+CYD=E, based on a matrix form of the algorithm LSQR for solving the least squares problem. We then apply this algorithm to compute the minimum norm least-squares centrosymmetric solution of min _X ‖AXB?E‖ _F . Numerical results are provided to verify the efficiency of the proposed method.     

A new solution to the generalized Sylvester matrix equation

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.