Topic
Riccati equation
About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.
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TL;DR: In this article, the Schrodinger equation is solved via a non-linear differential equation and the quantization of a quantum action is investigated in detail, which allows a rapid second-order convergence onto a desired eigenenergy En.
Abstract: Milne's approach to the numerical solution of the Schrodinger equation via a non-linear differential equation an the quantisation of a quantum action is investigated in detail. An accurate and efficient computational method is presented which allows a rapid second-order convergence onto a desired eigenenergy En. Numerical sample calculations demonstrate the efficiency of the method, which has special advantages for accurate calculations of high quantum states. The present method can be easily extended to the calculation of quasi-bound levels at resonance (complex-values) energies.
107 citations
TL;DR: In this paper, a kind of supersolutions of the so-called p-parabolic equation are studied, which are defined as lower semicontinuous functions obeying the comparison principle.
Abstract: A kind of supersolutions of the so-called p-parabolic equation are studied. These p-superparabolic functions are defined as lower semicontinuous functions obeying the comparison principle. Incidentally, they are precisely the viscosity supersolutions. One of our results guarantees the existence of a spatial Sobolev gradient. For p = 2 we have the supercaloric functions and the ordinary heat equation.
107 citations
TL;DR: It is shown that if there exists any stabilizing feedback leading to a Lyapunov function with star-convex level sets, then there always exists a representation of the dynamics such that the SDRE approach is stabilizing.
Abstract: The state-dependent Riccati equation (SDRE) approach to nonlinear system stabilization relies on representing a nonlinear system's dynamics in a manner to resemble linear dynamics, but with state-dependent coefficient matrices that can then be inserted into state-dependent Riccati equations to generate a feedback law. Although stability of the resulting closed-loop system need not be guaranteed a priori, simulation studies have shown that the method can often lead to suitable control laws. In this note, we consider the nonuniqueness of state-dependent representations. In particular, we show that if there exists any stabilizing feedback leading to a Lyapunov function with star-convex level sets, then there always exists a representation of the dynamics such that the SDRE approach is stabilizing. The main tool in the proof is a novel application of the S-procedure for quadratic forms.
106 citations
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TL;DR: In this paper, the authors considered the feedback system as a linear operator which maps the input w onto the output z. The most important norms for controller synthesis are the H 2 norm and the H ∞ norm.
Abstract: In Chap. 6 we studied the basic structure and fundamental properties of MIMO feedback systems. This chapter and the following two are concerned with controller synthesis. The idea is to consider the feedback system as a linear operator which maps the input w onto the output z. The feedback loop is given as in Sect. 6.5.1, which means in particular that the plant may contain weights. These weights formulate the design goals and the controller has to be constructed in such a way that it minimizes the operator norm of the feedback system. Operator norms of linear systems governed by ordinary differential equations were introduced in the previous chapter. The most important norms for controller synthesis are the H 2 norm and the H ∞ norm.
106 citations
TL;DR: The construction and analysis of a high order variant of the classical Chebyshev method that has order of convergence at least six for simple roots and an application to well-known algebraic Riccati equation is considered.
106 citations