2,879 research outputs found
Quantum alpha-determinants and q-deformed hypergeometric polynomials
The quantum -determinant is defined as a parametric deformation of
the quantum determinant. We investigate the cyclic
-submodules of the quantum matrix algebra
generated by the powers of the quantum
-determinant. For such a cyclic module, there exists a collection of
polynomials which describe the irreducible decomposition of it in the following
manner: (i) each polynomial corresponds to a certain irreducible
-module, (ii) the cyclic module contains an
irreducible submodule if the parameter is a root of the corresponding
polynomial. These polynomials are given as a -deformation of the
hypergeometric polynomials. This is a quantum analogue of the result obtained
in our previous work [K. Kimoto, S. Matsumoto and M. Wakayama,
Alpha-determinant cyclic modules and Jacobi polynomials, to appear in Trans.
Amer. Math. Soc.].Comment: 10 page
Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces
The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems
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