Abstract
Consider again the problem of optimal control of a process described by the following differential equation \(\dot{x}=f(x, u), \qquad x(0)=x_0, \quad u\in U\).
Who has taught us the true analogies, the
profound analogies which the eyes do not see,
but which reason can divine? It is the mathe-
mathical mind, which scorns content and clings
to pure form.
Henri Poincaré
“Analysis and Physics”
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Notes
- 1.
A piecewise smooth set \(M\subset G\) is a set which is a union of a finite or infinite number of curved polyhedrons from which only a finite number intersects with any bounded closed subset of G. The dimension of M is k if among the curved polyhedrons of which the sum is M there is a polyhedron of dimension k and the other polyhedrons have dimensions not greater than k.
A curved polyhedron in the space X is the image of a polyhedron in some finite dimensional vector space through a one-to-one smooth map.
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Górecki, H. (2018). Dynamic Programming [1, 2, 4, 5]. In: Optimization and Control of Dynamic Systems . Studies in Systems, Decision and Control, vol 107. Springer, Cham. https://doi.org/10.1007/978-3-319-62646-8_12
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