Non-Symmetric Matrix Riccati Equations

We prove a fundamental representation formula for all solutions of the matrix Riccati differential equation and of the corresponding algebraic Riccati equation. This formula contains the complete information on the phase portrait of the matrix equation and on the structure of the set F of all solutions of the corresponding algebraic equation. In particular we describe all constant, periodic and almost-periodic solutions of the matrix Riccati differential equation. Further we give an application of the fundamental representation formula to the investigation of non-autonomous Riccati equations.

[1]  G. Jank.  Asymptotic distribution of singularities of solutions of Matrix-Riccati differential equations , 1992, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[2]  Mark A. Shayman,et al.  Geometry of the Algebraic Riccati Equation, Part II , 1983 .

[3]  B. Anderson,et al.  Polynomial Factorization via the Riccati Equation , 1976 .

[4]  H. Meyer,et al.  The Matrix Equation $AZ + B - ZCZ - ZD = 0$ , 1976 .

[5]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[6]  On the variety of invariant subspaces of a finite-dimensional linear operator , 1982 .

[7]  R. Langer.  On the zeros of exponential sums and integrals , 1931 .

[8]  F. Callier,et al.  Criterion for the convergence of the solution of the Riccati differential equation , 1981 .

[9]  K. Mårtensson,et al.  On the matrix riccati equation , 1971, Inf. Sci..

[10]  R. Bucy.  Structural Stability for the Riccati Equation , 1975 .

[11]  M. Shayman.  Geometry of the Algebraic Riccati Equation, Part I , 1983 .

[12]  J. Potter.  Matrix Quadratic Solutions , 1966 .

[13]  H. Abou-Kandil,et al.  Necessary conditions for constant solutions of coupled Riccati equations in Nash games , 1993 .

[14]  K. Mrtensson.  On the matrix riccati equation , 1971 .

[15]  J. Medanic,et al.  Geometric properties and invariant manifolds of the Riccati equation , 1982 .

[16]  Werner Balser,et al.  Birkhoff invariants and stokes' multipliers for meromorphic linear differential equations , 1979 .

[17]  D. Zwillinger.  Matrix Riccati Equations , 1992 .

[18]  Mark A. Shayman,et al.  Phase portrait of the matrix Riccati equation , 1986 .