In this paper we consider various stability properties of real invariant lagrangian subspaces for real matrices which are either symmetric or skew-symmetric in a real quadratic form which may be symmetric or skew-symmetric itself. In particular, apart from ordinary stability we shall consider strong stability, which seems to be more desirable from a numerical point of view. For the classes of matrices we consider here stable subspaces are not always strongly stable, in contrast with the previous work. We shall completely characterize strongly stable invariant lagrangian subspaces, and in many cases also the stable ones. Invariant lagrangian subspaces with other stability properties, such as Lipschitz stability, are characterized as well.

Stability of Invariant Lagrangian Subspaces II

This review is concerned with two algebraic Riccati equations The first is a quadratic matrix equation for an unknown n × n matrix X of the form 

$$XDX + XA + A*X - C = 0,$$

(21)

where A, D, C are n × n complex matrices with C and D hermitian Further hypotheses are imposed as required, although Section 23 contains some discussion of more general non-symmetric quadratic equations The second equation has the fractional form 

$$X = A*XA + Q - (C + B*XA)*{(R + B*XB)^{ - 1}}(C + B*XA),$$

(22)

where R and Q are hermitian m × m and n × n matrices, respectively, and A, B, C are complex matrices with respective sizes n × n, n × m, and m × n The two equations are frequently referred to as the “continuous” and “discrete” Riccati equations, respectively, because they arise in physical optimal control problems in which the time is treated as a continuous variable, or a discrete variable

Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review

This paper deals with the discrete-time infinite-horizon linear quadratic problem with indefinite cost criterion. Given a discrete-time linear system, an indefinite cost-functional and a linear subspace of the state space, we consider the problem of minimizing the cost-functional over all inputs that force the state trajectory to converge to the given subspace. We give a geometric characterization of the set of all hermitian solutions of the discrete-time algebraic Riccati equation. This characterization forms the discrete-time counterpart of the well-known geometric characterization of the set of all real symmetric solutions of the continuous-time algebraic Riccati equation as developed by Willems [IEEE Trans. Automat. Control, 16 (1971), pp. 621- 634] and Coppel [Bull. Austral. Math. Soc., 10 (1974), pp. 377-401]. In the set of all hermitian solutions of the Riccati equation we identify the solution that leads to the optimal cost for the above mentioned linear quadratic problem. Finally, we give necessary and sufficient conditions for the existence of optimal controls. Keywords: Discrete time optimal control, indefinite cost, algebraic Riccati equation, linear endpoint constraints.

Linear quadratic problems with indefinite cost for discrete time systems

Stable Invariant Lagrangian Subspaces: Factorization of Symmetric Rational Matrix Functions and Other Applications

Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle. We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability as well as the index of stability for these subspaces.

Perturbation analysis of Lagrangian invariant subspaces of symplectic matrices

The stability of various factorizations of self-adjoint rational matrix functions and matrix polynomials, as well as of hermitian solutions of symmetric matrix algebraic Riccati equations, is studied. In the first part of this paper results on stability of certain classes of invariant subspaces of a matrix which is self-adjoint in an indefinite inner product were obtained. These results serve as the main tools in the investigation.

Algebraic Riccati Equations

Papers

Stability of Invariant Maximal Semidefinite Subspaces. II. Applications: Self-Adjoint Rational Matrix Functions,