Riccati Differential Equations

Reid, William J.; Chidambara, M. R.

doi:10.1109/tsmc.1977.4309812

Cited by 115 publications

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“…Equations like those above for the second-order moments of the conditioned state arise very frequently in classical continuous-time observation and control problems. They can be collected into a Riccati matrix differential equation [28] for the covariance matrix,…”

Section: A a Generalized Model For Continous Position Measurementmentioning

confidence: 99%

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State determination in continuous measurement

et al. 1999

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The possibility of determining the state of a quantum system after a continuous measurement of position is discussed in the framework of quantum trajectory theory. Initial lack of knowledge of the system and external noises are accounted for by considering the evolution of conditioned density matrices under a stochastic master equation. It is shown that after a finite time the state of the system is a pure state and can be inferred from the measurement record alone. The relation to emerging possibilities for the continuous experimental observation of single quanta, as for example in cavity quantum electrodynamics, is discussed.42.50. Lc,03.65.Bz,42.50.Ct

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Section: A a Generalized Model For Continous Position Measurementmentioning

confidence: 99%

“…The reason for this is clear in this context, such an unravelling corresponds to a measurement in which the observer obtains no information about the system state. The matrix Riccati equation (2.5a) has an analytic solution given by Reid [28]. Where U (t) and W (t) obey the linear coupled matrix equations…”

Section: B Steady State Conditioned Variancesmentioning

confidence: 99%

State determination in continuous measurement

et al. 1999

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“…In this section, we present several applications of the global oscillation theorem (Theorem 3.5). For a given λ ∈ R, consider the quadratic functional Other conditions equivalent to the positivity and nonnegativity of F(·, λ) are well-known in the literature, such as the solvability of Riccati matrix equations and inequalities or perturbations of the boundary conditions of admissible pairs, see e.g., [4], [6], [14]. Next, we introduce another useful tool from the theory of quadratic functionals.…”

Section: Applicationsmentioning

confidence: 99%

Oscillation and spectral theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

Böhner

Kratz

Hilscher

2012

Mathematische Nachrichten

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Key words Linear Hamiltonian system, self-adjoint eigenvalue problem, proper focal point, conjoined basis, finite eigenvalue, oscillation, controllability, normality, quadratic functional MSC (2010) 34L05, 34C10, 49N10, 93B60, 34L10In this paper, we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. Our results generalize the known theory of linear Hamiltonian systems in two respects. Namely, we allow nonlinear dependence of the coefficients on the spectral parameter and at the same time we do not impose any controllability and strict normality assumptions. We introduce the notion of a finite eigenvalue and prove the oscillation theorem relating the number of finite eigenvalues which are less than or equal to a given value of the spectral parameter with the number of proper focal points of the principal solution of the system in the considered interval. We also define the corresponding geometric multiplicity of finite eigenvalues in terms of finite eigenfunctions and prove that the algebraic and geometric multiplicities coincide. The results are also new for Sturm-Liouville differential equations, being special linear Hamiltonian systems.The coefficients are n × n-matrix-valued functions such that the Hamiltoniandefined on [a, b] × R is symmetric, satisfies certain smoothness assumptions (see Section 2), and the monotonicity assumption

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“…(2) Differential nonsymmetric Riccati equations (NDREs) play a fundamental role in many areas such as transport theory, fluid queues models, variational theory, optimal control and filtering, H 1 -control, invariant embedding and scattering processes, dynamic programming, and differential games. [1][2][3][4][5] For NAREs, many numerical methods have been studied for finding the minimal nonnegative solution X * . The Newton method has been studied in other works 2,6,7 ; however, because it requires at each step the solution of a Sylvester equation, the method could be expensive when direct solvers are used.…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

Angelova

Hached

Jbilou

2019

Numerical Linear Algebra App

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Summary In this paper, we consider large‐scale nonsymmetric differential matrix Riccati equations with low‐rank right‐hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability, and others. We show how to apply Krylov‐type methods such as the extended block Arnoldi algorithm to get low‐rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as backward differentiation formula (BDF) or Rosenbrock method. We also show how these techniques can be easily used to solve some problems from the well‐known transport equation. Some numerical examples are given to illustrate the application of the proposed methods to large‐scale problems.

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Riccati Differential Equations

Cited by 115 publications

References 0 publications

State determination in continuous measurement

State determination in continuous measurement

Oscillation and spectral theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

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