Approximate solutions with a priori error bounds for continuous coefficient matrix Riccati equations

In this paper, the exact solution of the nonsymmetric matrix Riccati equation with continuous coefficients is approximated using Fer's approximations of the associated underlying linear system. Given an admissible error @e > 0, the order n of Fer's truncation is determined so that in the previously guaranteed existence interval, the error of the approximated solution is less than @e. Some qualitative properties of the Fer's approximations are given and illustrative examples are included.

[1]  W. Reid,et al.  Riccati Differential Equations , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[2]  John L. Casti,et al.  Dynamical Systems and Their Applications : Linear Theory , 1977 .

[3]  R. Wilcox.  Exponential Operators and Parameter Differentiation in Quantum Physics , 1967 .

[4]  C. Kenney,et al.  Numerical integration of the differential matrix Riccati equation , 1985 .

[5]  Marianela Lentini,et al.  INVARIANT IMBEDDING, THE BOX SCHEME AND AN EQUIVALENCE BETWEEN THEM* , 1982 .

[6]  Enrique Ponsoda,et al.  Non-autonomous Riccati-type matrix differential equations: existence interval, construction of continuous numerical solutions and error bounds , 1995 .

[7]  A. Iserles,et al.  On the solution of linear differential equations in Lie groups , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  L Jódar,et al.  Rational matrix approximation with a priori error bounds for non-symmetric matrix Riccati equations with analytic coefficients , 1998 .

[9]  D. Evelyne.  Thèse de doctorat d'Etat , 1988 .

[10]  Luca Dieci,et al.  Numerical integration of the differential Riccati equation and some related issues , 1992 .

[11]  Jeremy Schiff,et al.  A Natural Approach to the Numerical Integration of Riccati Differential Equations , 1999 .

[12]  Elena Celledoni,et al.  Approximating the exponential from a Lie algebra to a Lie group , 2000, Math. Comput..

[13]  S. Clemhout,et al.  Interactive economic dynamics and differential games , 1979 .

[14]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[15]  A. J. Laub,et al.  Efficient matrix-valued algorithms for solving stiff Riccati differential equations , 1990 .

[16]  Yaakov Oshman,et al.  Eigenfactor solution of the matrix Riccati equation - A continuous square root algorithm , 1984 .

[17]  R. Feynman.  An Operator calculus having applications in quantum electrodynamics , 1951 .

[18]  Roger B. Sidje,et al.  Expokit: a software package for computing matrix exponentials , 1998, TOMS.

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  Arieh Iserles,et al.  Solving linear ordinary differential equations by exponentials of iterated commutators , 1984 .

[21]  Thomas Kailath,et al.  Linear Systems , 1980 .

[22]  H. Abou-Kandil,et al.  A resolution method for Riccati differential systems coupled in their quadratic terms , 1988 .

[23]  Arunabha Bagchi.  Stackelberg Differential Games in Economic Models , 1984 .

[24]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[25]  J. B. Cruz,et al.  Series Nash solution of two-person, nonzero-sum, linear-quadratic differential games , 1971 .

[26]  R. Meire,et al.  The stability of the triangular points in the elliptic restricted problem , 1981 .

[27]  Sergio Blanes,et al.  Continuous numerical solutions of coupled mixed partial differential systems using Fer's factorization , 1999 .

[28]  Volker Mehrmann,et al.  Numerical solution of differential algebraic riccati equations , 1990 .

[29]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .