A computational solution for a Matrix Riccati differential equation

Summary This paper is concerned with the solution of the finite time Riccati equation. The solution to the Riccati equation is given in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and are solved using computational methods. Examples illustrating the method are presented and the computational algorithms are given.     

Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

We consider the initial value problem for a nonsymmetric matrix Riccati differential equation, where the four coefficient matrices form an M-matrix. We show that for a wide range of initial values the Riccati differential equation has a global solution X(t) on [0,∞) and X(t) converges to the stable equilibrium solution as t goes to infinity.     

On the solution of a class of the reduced wave equation and the Riccati differential equation

In this paper we present an “iteration” technique for a class of differential equation having the form z^″=λz, where λ is a function in C^∞. We show that we can construct not only the general solution of the reduced wave equation but also the general solution of the Riccati differential equation by using this iteration technique if the given function λ is satisfies the condition

     

Conservation of a bounded solution of the Riccati equation under a small perturbation

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 7, pp. 965–969, July, 1989.     

Time-optimal solution of a linear evolution equation in Banach spaces

The problem is that of finding trajectories of a linear evolution equation connecting two prescribed sets of states, initial and terminal, in the shortest possible time. Necessary conditions for the existence of solutions, i.e., time-optimum trajectories, are given in the form of a maximum principle.     

Infinite products in a Banach algebra

A theory is developed for infinite products in a noncommutative Banach algebra. Sufficient conditions for the convergence of such a product are given. Conditions for analyticity of the product are also given in the case when the factors depend on a complex parameter.     

On the S-transform over a Banach algebra

The S-transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B-valued Banach noncommutative probability space, for an arbitrary unital complex Banach algebra B. Also, a new proof of the additivity of the R-transform in this setting is given.     

New upper solution bounds of the discrete algebraic Riccati matrix equation

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.     

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If \frak A {\frak A} is a reflexive, amenable Banach algebra such that for each maximal left ideal L of \frak A {\frak A} (i) the quotient \frak A/L {\frak A}/L has the approximation property and (ii) the canonical map from \frak A \check? L^{^} {\frak A} \check{\otimes} L^\perp to (\frak A / L) \check? L^{^} ({\frak A} / L) \check{\otimes} L^\perp is open, then \frak A {\frak A} is finite-dimensional. As an application, we show that, if \frak A {\frak A} is an amenable Banach algebra whose underlying Banach space is an \scr L^p {\scr L}^p -space with p ? (1,￥) p\in (1,\infty) such that for each maximal left ideal L the quotient \frak A/L {\frak A}/L has the approximation property, then \frak A {\frak A} is finite-dimensional.     

Exterior algebra of a Banach space

As far as we know, the exterior product with any norm has not been studied for Banach spaces. Especially, no studies have been done on Grassmann manifolds in Banach spaces. We think it is important to study these because simple m-vectors can be thought of as m-dimensional subspaces scaled in some way according to our work. We hope Banach space norms of simple m-vectors will yield metric information about their associated subspaces. In fact, this is the case with m-uniform convexity and m-uniform rotundity which are associated with area (in Banach spaces).     

The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space

We consider linear fractional differential operator equations involving the Caputo derivative. The goal of this paper is to establish conditions for the unique solvability of the inverse Cauchy problem for these equations. We use properties of the Mittag-Leffler function and the calculus of sectorial operators in a Banach space. For equations with operators in a general form we obtain sufficient conditions for the unique solvability, and for equations with densely defined sectorial operators we obtain necessary and sufficient unique solvability conditions.     

On the approximation of a bounded solution of a linear differential equation in a Banach space

We investigate the problem of approximation of a bounded solution of a linear differential equation by solutions of the corresponding difference equations in a Banach space. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1268–1271, September, 1998.