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2 edition of On positive semidefinite solutions of the operator Lyapunov equation. found in the catalog.

On positive semidefinite solutions of the operator Lyapunov equation.

Henrik Stetkaer

# On positive semidefinite solutions of the operator Lyapunov equation.

## by Henrik Stetkaer

Written in English

Edition Notes

The Physical Object ID Numbers Series Preprint series -- 1975/76, No. 31. Pagination 1 bd. (flere p.agineringer) Open Library OL21081823M

• then there exists a Lyapunov function that proves it a sharper converse Lyapunov theorem is more speciﬁc about the form of the Lyapunov function example: if the linear system x˙ = Ax is G.A.S., then there is a quadratic Lyapunov function that proves it (we’ll prove this later) Basic Lyapunov . () Symmetric, positive semidefinite, and positive definite real solutions of AX = XAT and AX = YB. Linear Algebra and its Applications , () A general analysis of Sylvester's matrix by:

unique positive semidefinite solutions of coupled Lyapunov equations corresponding to discrete- time jump linear systems if they exist. The algorithm has the advantage that it operates on reduced-order decoupled equations, thus allowing for parallel processing. In addition, the com-. along the system trajectories. Now suppose that there exists a positive semidefinite matrix Q such that AP PA QT +=− Then 11 22() 0 VxAPPAx xQx =+=−≤TT T Then, V(x) is a Lyapunov function and the system is SISL. We call AP PA QT +=− the continuous-time Lyapunov Size: KB.

SOLUTION OF THE LYAPUNOV EQUATION where cu and xn are scalars and s, c and x are (n-1) element vectors. Then Equation () gives the three equations (AI+XJXH = -cn and hence () () Once xu has been found from Equation (), Equation () can be solved, by forward substitution, for x and then Equation () is of the same form as (), but of. The main result concerning the Sylvester equation is the following: If and have no common eigenvalues, then the Sylvester equation has a unique solution for any.. When and there are no eigenvalues of such that whatever and are (in the numbering of eigenvalues of), then (a1) has a unique Hermitian solution for er if is a Hurwitz matrix (i.e. having all its eigenvalues in the left.

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### On positive semidefinite solutions of the operator Lyapunov equation by Henrik Stetkaer Download PDF EPUB FB2

The case where C is positive semidefinite but where the space is finite dimensional, i.e. where (1) is the matrix Lyapunov equation, has been treated in  and . The new feature of this paper is a treatment of positive semidefinite solutions to the Lyapunov equation (1) Cited by: 2.

The new feature of this paper is a treatment of positive semidefinite solutions to the Lyapunov equation (1) in a complex Hilbert space S which in general is infinite dimensional. Our main results are (1) The equation (1) has a positive semidefinite bounded solution if and only if the limit.

() On positive semidefinite solutions of the operator Lyapunov equation. Journal of Mathematical Analysis and Applications() Controllability and inertia theory for functions of a by: In this paper, we present three iterative algorithms for symmetric positive semidefinite solutions of the Lyapunov matrix equations.

The first and second iterative algorithms are based on the relaxed proximal point algorithm (RPPA) and the Peaceman&#x;Rachford splitting method (PRSM), respectively, and their global convergence can be ensured by corresponding results in the : Min Sun, Jing Liu.

is Lyapunov stable (see e.g. ). This operator equation also arises in linear quadratic differential games (e.g. ). The purpose of this note is to show that there is a general abstract setting for the solution which is independent of dimensionality and also the specific context in which the equation Author: Richard Datko.

In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others. () Positive Solutions of a Class of Operator Equations.

Ukrainian Mathematical Journal() On the eigenvalue estimation for solution to Lyapunov by: Let A be a positive definite operator and B be a positive semidefinite operator. There exists positive semidefinite operator solution X of the following operator equation (i), (ii), (iii), (iv) and (v) respectively: (i) A 2+r 2 X + XA 2+r 2 = A r 2 (AB + BA)A r 2 for r by: 2.

The aim of this paper is to investigate the existence and uniqueness of solutions for a class of fractional q-difference Schrödinger equations appeared in Li et al. (Appl Math Lett – Analytic solution. Defining the operator as stacking the columns of a matrix and as the Kronecker product of and, the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation.

Furthermore, if the matrix is stable, the solution can also be expressed as an integral (continuous time case). Semidefinite solutions to Lyapunov equation. Ask Question Asked 4 years, Browse other questions tagged linear-algebra matrices matrix-equations positive-semidefinite or ask your own question.

Identify a book where the main character is released from prison in order to be sent to conduct espionage in Italy. Consider the Lyapunov equation (). We shall henceforth assume that the matrix A is stable, that Q = Qr i> o, and that (A, Q) is a controllable pair. Under these conditions, the solution X of the Lyapunov equation is positive semidefinite.

Positive definite solutions of some matrix equations. In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others.

In order to investigate the stability of a system using this method, a positive energy-like function of the states on the trajectory of the differential equation is taken into account, which is called the Lyapunov function.

A Lyapunov function decreases along the trajectory of the ODE. Positive semidefinite solutions of the operator equation Sigma(n)(j=1)A(n-j)XA(j-1) = B Article in Linear Algebra and its Applications (4) February with 9 Reads.

• if A is stable, Lyapunov operator is nonsingular • if A has imaginary (nonzero, iω-axis) eigenvalue, then Lyapunov operator is singular thus if A is stable, for any Q there is exactly one solution P of Lyapunov equation ATP +PA+Q = 0 Linear quadratic Lyapunov theory 13–7.

Integral (sum) solution of Lyapunov equation If x_ = Ax is (globally asymptotically) stable and Q = QT, P = Z 1 0 eATtQeAt dt is the unique solution of the Lyapunov equation ATP +PA+Q = 0.

If x(t+1) = Ax(t) is (globally asymptotically) stable and Q = QT, P = X1 t=0 (AT)tQAt is the unique solution of the Lyapunov equation ATPA P +Q = 0. This paper deals with the problems of eigenvalue estimation for the solution to the perturbed matrix Lyapunov equation.

We obtain some eigenvalue inequalities on condition that X is a positive. The analysis covers both the causal and noncausal cases. In particular, the asymptotic stability of a discrete descriptor system (DDS) is related to the existence of a positive semidefinite solution of the generalized Lyapunov equation.

The results strengthened those Author: Qingling Zhang, James Lam, Liqian Zhang. The above equation admits a unique symmetric positive semidefinite solution X. Thus, such a solution matrix X has the Cholesky factorization X = YTY, where Y is upper triangular. In several applications, all that is needed is the matrix Y; X is not needed as such.

Lyapunov’s first method requires the solution of the differential equations describing the dynamics of the system which makes it impractical in the analysis and design of control systems.

As will be seen later, Eq. (5) reduces to the observability Lyapunov equation upon the choice Q = C T C, where C is the output matrix of the dynamic system.For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions.

We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite by: Linear algebra restatement of the matrix deﬁnition.

An operator Ais positive deﬁnite if for all nonzero x, xAx>0. Consider f(x) on [0,1],suchthatf(0) = f(1) = 0. The set of such functions is a linear subspace. Deﬁne the dot product according to fg= Z 1 0 f(x)g(x)dx.

Consider the linear operator Athat is taking the second derivative of File Size: KB.