### Control of Complex Nonlinear Systems with Delay

First order ordinary differential equations are often exactly solvable by separation of variables , especially for autonomous equations. For example, the nonlinear equation. The equation is nonlinear because it may be written as. Note that if the u 2 term were replaced with u , the problem would be linear the exponential decay problem. Second and higher order ordinary differential equations more generally, systems of nonlinear equations rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered. Common methods for the qualitative analysis of nonlinear ordinary differential equations include:.

The most common basic approach to studying nonlinear partial differential equations is to change the variables or otherwise transform the problem so that the resulting problem is simpler possibly even linear. Sometimes, the equation may be transformed into one or more ordinary differential equations , as seen in separation of variables , which is always useful whether or not the resulting ordinary differential equation s is solvable. Another common though less mathematic tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem.

For example, the very nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation. Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.

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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics , it may be shown  that the motion of a pendulum can be described by the dimensionless nonlinear equation. Another way to approach the problem is to linearize any nonlinearities the sine function term in this case at the various points of interest through Taylor expansions.

This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state. This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. sketching phase portraits

March Learn how and when to remove this template message. Main articles: Algebraic equation and System of polynomial equations. See also: List of nonlinear partial differential equations.

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Main article: Pendulum mathematics. MIT News. Retrieved March Annals of Biomedical Engineering. Bibcode : Nonli.. Bibcode : Chaos.. Bibcode : NatSR Berlin: Springer. Retrieved 20 January Bibcode : Natur. Journal of Symbolic Computation.

## Control of Complex Nonlinear Systems with Delay

Diederich Hinrichsen and Anthony J. Pritchard Springer Verlag.

Jordan, D. Nonlinear Ordinary Differential Equations fourth ed. Oxford University Press. Khalil, Hassan K.

## Control of Complex Nonlinear Systems with Delay | Philipp Hövel | Springer

Nonlinear Systems. Prentice Hall. Kreyszig, Erwin Advanced Engineering Mathematics. Volume , Issue Previous Article. Technical Briefs. This Site. Google Scholar. Shubhendu Bhasin Shubhendu Bhasin.

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