Abstract
In this paper we develop new theory of Riccati matrix differential equations for linear Hamiltonian systems, which do not require any controllability assumption. When the system is nonoscillatory, it is known from our previous work that conjoined bases of the system with eventually the same image form a special structure called a genus. We show that for every such a genus there is an associated Riccati equation. We study the properties of symmetric solutions of these Riccati equations and their connection with conjoined bases of the system. For a given genus, we pay a special attention to distinguished solutions at infinity of the associated Riccati equation and their relationship with the principal solutions at infinity of the system in the considered genus. We show the uniqueness of the distinguished solution at infinity of the Riccati equation corresponding to the minimal genus. This study essentially extends and completes the work of W. T. Reid (1964, 1972), W. A. Coppel (1971), P. Hartman (1964), W. Kratz (1995), and other authors who considered the Riccati equation and its distinguished solution at infinity for invertible conjoined bases, i.e., for the maximal genus in our setting.
Highlights
Riccati differential equations for self-adjoint linear differential systems play fundamental role in mathematical research as well as in applications
We showed in [28, 29] the existence of principal solutions (X, U ) at infinity with all ranks of X (t) in a specific range depending on the maximal order of abnormality d∞ of (H), their classification and limit properties with antiprincipal solutions at infinity [30], and the geometric structure of the set of all conjoined bases [31]
(iii) we show that every such a Riccati equation (R) possesses a distinguished solution at infinity, which corresponds to a principal solution of (H) at infinity from the genus G
Summary
Riccati differential equations for self-adjoint linear differential systems play fundamental role in mathematical research as well as in applications. If n ∈ N is a given dimension and A, B, C : [a, ∞) → Rn×n are given piecewise continuous matrix-valued functions such that B(t) and C(t) are symmetric, the Riccati matrix differential equation. The Riccati equation (R) has many applications in various disciplines, for example in the oscillation and spectral theory [2, 7, 17, 22, 23, 24], filtering and prediction theory [16, 23], calculus of variations and optimal control theory [1, 3, 8, 12, 10, 2010 Mathematics Subject Classification. Linear Hamiltonian system, Riccati differential equation, genus of conjoined bases, distinguished solution at infinity, principal solution at infinity, controllability.
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