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Invariant imbedding and Riccati transformations
Applied Mathematics and Computation, 1975From its inception, the theory of invariant imbedding has been concerned with the study of various relations between the inputs and outputs of various physical processes. Where the processes could be modelled by differential or integro-differential equations, these ideas have led to the heuristic development of various functional relationships for the ...
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Decoupling Order Reduction via the Riccati Transformation
SIAM Review, 1987A program that dates back to Riccati [1724] and that seeks to replace a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use of suitable transformations has seen a notable advance in the last twenty years.
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Reports on Mathematical Physics, 2005
The author asks the following simple question: given two functions \(a, b\in C^{1}(\mathbb{R},\mathbb{R}),\) and \(n\geq 2,\) when can we solve by quadrature the generalized Riccati-Abel equation \(y^{\prime }=y^{n}+a(x)y+b(x)?\) For the simple case \(n=2\) and \(a=0,\) the problem goes back to Liouville, where methods of differential algebra were ...
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The author asks the following simple question: given two functions \(a, b\in C^{1}(\mathbb{R},\mathbb{R}),\) and \(n\geq 2,\) when can we solve by quadrature the generalized Riccati-Abel equation \(y^{\prime }=y^{n}+a(x)y+b(x)?\) For the simple case \(n=2\) and \(a=0,\) the problem goes back to Liouville, where methods of differential algebra were ...
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Transforming algebraic Riccati equations into unilateral quadratic matrix equations
Numerische Mathematik, 2010Given \(A\in\mathbb{R}^{m\times m}\), \(B\in\mathbb{R}^{m\times n}\), \(C\in\mathbb{R}^{n\times m}\) and \(D\in\mathbb{R}^{n\times n}\), the authors reduce the nonsymmetric algebraic Riccati equation (ARE), \(XCX-AX-XD+B=0\), where \(X\in\mathbb{R}^{m\times n}\) is the unknown, to a unitaral quadratic matrix equation (UQME), \(\mathcal{P}\mathcal{X}^2+\
BINI, DARIO ANDREA +2 more
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Riccati transformations for control optimization using the second variation
1970 IEEE Symposium on Adaptive Processes (9th) Decision and Control, 1970Two new numerical methods which may be used to calculate solutions to optimal control problems are developed. These methods involve guessing initial values for unknown Lagrange multipliers and a control sequence. They are similar to the successive sweep method in that Riccati equations are used to calculate Corrections to these guessed variables.
Williamson, Walt E., Tapley, Byron D.
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On linearization of Riccati differential equations through variable transformations
2008 International Conference on Control, Automation and Systems, 2008Exact linearization of a Riccati differential equation, which has a stable equilibrium and an unstable one, is considered using variable transformations. Of particular interests are what happens to equilibrium points of the nonlinear equation when it is linearized and how these points are related to linearizing transformation.
null Tsubasa Kittaka, Noriyuki Hori
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A Useful Fractional Linear Transformation for the Riccati Equation for Reflection Coefficients
Microwave and Optical Technology Letters, 1995AbstractThe conventional Riccati equation for the reflection coefficient involves the spatial derivative of the refractive index, and thus it is not suitable for use when the refractive index is nondifferentiable and/or discontinuous. In the present article we introduce a fractional linear transformation to obtain a Riccati equation that does not ...
Sailing He
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Möbius Transformations and Periodic Solutions of Complex Riccati Equations
Bulletin of the London Mathematical Society, 1997The paper studies the solutions of a complex \(T\)-periodic Riccati equation \[ Z'= Z^2+p(t),\tag{\(*\)} \] including the possible dynamics for this type of equation. It is investigated on the Riemann sphere, \(\widehat{\mathbb{C}}= \mathbb{C}\cup\{\infty\}\), where \(p:\mathbb{R}/T\mathbb{Z}\mapsto \mathbb{C}\) is a continuous function.
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On the Oscillation of Fourth-Order Delay Differential Equations via Riccati Transformation
MathematicsThis paper deals with the oscillatory behavior of solutions of a general class of fourth-order non-linear delay differential equations. New oscillation criteria are established using Riccati transformation and a Philos-type technique.
Mohamed M. A. El-Sheikh +3 more
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Modal factorization of time-varying models for nonlinear circuits by the Riccati transform
ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196), 2002In this paper, general linear time-varying systems are addressed. They are considered as small-signal models of nonlinear circuit solutions. The transition matrix is constructed by repeated Riccati transforms. It is shown that each transform factors out a single mode.
Pieter van der Kloet, Fred L. Neerhoff
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