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A note about this: much of the optimal control theory work surrounding Pontrayagin's Principle are theoretical building blocks whose primary utility is for math
by wenc 8y ago
A note about this: much of the optimal control theory work surrounding Pontrayagin's Principle are theoretical building blocks whose primary utility is for mathematical analysis.
In practice, they result in boundary-value problems that are impractical to solve for systems of non-trivial size, and are almost never implemented in practice. Occasionally they are used to construct parameterized solutions for extremum control (e.g. NCO tracking) for very small systems, but these tend to be rarer cases. One runs into dimensionality issues very quickly.
In industrial control systems, optimal control models are almost always discretized and the optimization is done on algebraic systems of equations. Linear algebra dominates there.
- westoncb 8y ago> ... are theoretical building blocks whose primary utility is for mathematical analysis. When you say 'mathematical analysis' here, is that just a broader category that encompasses e.g. real and complex analysis? Or something else?
- whatshisface 8y agoIndustry usually means "using math to analyze" when they say "mathematical analysis."
- wenc 8y agoYes, that is what I meant [1]. For example, you can use ideas from differential equations/analysis to determine say, the existence and uniqueness of solutions for continuous ODEs. Pontryagin's Principle and the calculus of variations in general gives you theoretical machinery for working with models in analytic form. Some problems such as minimum time optimization are more tractable in continuous time form than in discrete time. You can also in some simple cases derive the set of closed-form optimal solution trajectories (unconstrained case) and analyze that directly. Once the models are transformed into discrete form for numerical solution, the tools used lie more in the realm of linear algebra (positive definiteness of Hessians, etc.) [1] https://en.wikipedia.org/wiki/Mathematical_analysis https://en.wikipedia.org/wiki/Mathematical_analysis