Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+AtX-1=I

Matrices;numerieke wiskunde

Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A*X-1A=Q

Matrices;numerieke wiskunde

A matricial boundary value problem which appears in the transport theory

AbstractWe study the boundary value problem Tφ′ = −Aφ (I– P)φ(0), Pφ(τ) = 0, where T = T∗, A = A∗, P2 = P, P∗T = TP, T invertible. The motivation to consider such problem comes from the transport theory. The behavior of values of τ for which there is a nontrivial solution (exceptional values) is investigated using the indicator function. This is an analytic hermitian valued function of the real parameter t which reflects, in particular, the characteristic properties of exceptional values

Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems

AbstractWe discuss two comparison theorems for algebraic Riccati equations of the form XBR-1 B∗X−X(A−BR-1C)-(A−BR-1C)∗X−(Q−C∗R-1C) = 0. Simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a Riccati equation from the existence of a hermitian solution of a second Riccati equation. Further, similar results are given for discrete algebraic Riccati equations

Linear quadratic problems with indefinite cost for discrete time systems

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

Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + A*X-1A = Q

AbstractWe consider the problem of when the matrix equation X + A∗X-1A = Q has a positive definite solution. Here Q is positive definite. We study both the real and the complex case. This equation plays a crucial role in solving a special case of the discrete-time Riccati equation. We present both necessary and sufficient conditions for its solvability. This result is obtained by using an analytic factorization approach. Moreover, we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation, respectively. Finally, we discuss the number of solutions

Nonnegative solutions of algebraic Riccati equations

AbstractNonnegative Hermitian solutions of various types of continuous and discrete algebraic Riccati equations are studied. The Hamiltonian is considered with respect to two different indefinite scalar products. For the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated. For general Hermitian solutions a method to compute the inertia is given. Although most attention is payed to the classical types arising from LQ optimal control theory, the case where the quadratic term has an indefinite coefficient is studied as well