On the commutative class of linear time-varying systems

Wu, Min-Yen; Sherif, Abdulaziz

doi:10.1080/00207177608922171

Cited by 33 publications

(8 citation statements)

References 2 publications

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“…For the time varying case, in general a computable expression of the transition state matrix <&(t,s) is not available, though there exist several classes of systems such that this matrix is computable in the same manner as the linear invariant systems. In accordance with Definition 2 given by Wu in [15] Proof. From Theorem 1, the only solution of a Cauchy problem for the differential equation of (1.1) is given by (2.3), where F = X(b).…”

Section: Explicit Solutionsmentioning

confidence: 63%

“…, a set of m x m constant matrices {Mi}p i=l is said to be mutually commutative if and only if, M i M j = M j M i , for all i,j, such that Let us consider the time-varying Problem (1.1) and let us suppose that the matrix M(t) given by (2.1) can be written as (2.11) where M,'s are constant matrices and c ; (t)'s are linearly independent sets of scalar time functions extracted from elements of M(t), such that {Mi}f =l is mutually commutative, then the transition state matrix <t(t, s) of the linear system (2.7) can be computed by (D(t,b)=nexp(MA(£,fc)) (2.12) where bi(t,b)=\ci(s)ds (2.13) b and M,-'s and c,(s)'s are defined by (2.11).The following corollary is a consequence of Theorem 1 and Theorems 1 and 2 of[15].…”

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confidence: 92%

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Explicit solutions for a system of coupled Lyapunov differential matrix equations

Jódar

Mariton

1987

Proceedings of the Edinburgh Mathematical Society

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This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the typewhere Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

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Section: Explicit Solutionsmentioning

confidence: 63%

mentioning

confidence: 92%

Explicit solutions for a system of coupled Lyapunov differential matrix equations

Jódar

Mariton

1987

Proceedings of the Edinburgh Mathematical Society

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“…However, the matrices A(t), t ∈ R are in companion form and are not in general commutative (see also Proposition 4.18). For further insight into the commutativity of some matrices A(t) and t t 0 A(s)ds as well as A(s) and A(t), we refer to [32,Exercise 4.8] and [41].…”

Section: Remark 44 If A(s) and A(t) Commute Ie [A(s) A(t)]mentioning

confidence: 99%

Continuous-time locally stationary time series models

Bitter¹,

Stelzer²,

Ströh³

2021

Preprint

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We adapt the classical definition of locally stationary processes in discrete-time (see e.g. [14]) to the continuous-time setting and obtain equivalent representations in the time and frequency domain. From this, a unique time-varying spectral density is derived using the Wigner-Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

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“…In addition, it holds [A(t), A(s)] = 0 for all s, t ∈ R and τ ∈ T (see e.g. [33]). All eigenvalues of A(t) are real and strictly negative for a properly restricted T .…”

Section: Assumption (C7) For Eachmentioning

confidence: 99%

“…where V ϑ , Ỹ ϑ * (u),∆ and ε ϑ are defined as in Section 5.2. Note that L(ϑ) is closely related to the limiting function of the quasi maximum likelihood estimator from Section 5.2 such that Proposition 5.6 along with the inequalities in (33) ensure that W is continuous. Moreover, Proposition 5.11 ensures that W (ϑ) has a unique minimum.…”

mentioning

confidence: 99%

Statistical inference for continuous-time locally stationary processes using stationary approximations

Ströh¹

2021

Preprint

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We establish asymptotic properties of M -estimators, defined in terms of a contrast function and observations from a continuous-time locally stationary process. Using the stationary approximation of the sequence, θ-weak dependence, and hereditary properties, we give sufficient conditions on the contrast function that ensure consistency and asymptotic normality of the M -estimator.As an example, we obtain consistency and asymptotic normality of a localized least squares estimator for observations from a sequence of time-varying Lévy-driven Ornstein-Uhlenbeck processes. Furthermore, for a sequence of time-varying Lévy-driven state space models, we show consistency of a localized Whittle estimator and an M -estimator that is based on a quasi maximum likelihood contrast. Simulation studies show the applicability of the estimation procedures.

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On the commutative class of linear time-varying systems

Cited by 33 publications

References 2 publications

Explicit solutions for a system of coupled Lyapunov differential matrix equations

Explicit solutions for a system of coupled Lyapunov differential matrix equations

Continuous-time locally stationary time series models

Statistical inference for continuous-time locally stationary processes using stationary approximations

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