By Magnus Egerstedt

Splines, either interpolatory and smoothing, have a protracted and wealthy historical past that has principally been program pushed. This ebook unifies those buildings in a accomplished and obtainable manner, drawing from the newest tools and purposes to teach how they come up certainly within the conception of linear keep an eye on structures. Magnus Egerstedt and Clyde Martin are prime innovators within the use of keep watch over theoretic splines to compile many diversified purposes inside a standard framework. during this ebook, they start with a sequence of difficulties starting from direction making plans to stats to approximation. utilizing the instruments of optimization over vector areas, Egerstedt and Martin exhibit how all of those difficulties are a part of a similar common mathematical framework, and the way they're all, to a undeniable measure, a end result of the optimization challenge of discovering the shortest distance from some degree to an affine subspace in a Hilbert house. They disguise periodic splines, monotone splines, and splines with inequality constraints, and clarify how any finite variety of linear constraints should be additional. This ebook unearths how the various typical connections among regulate conception, numerical research, and information can be utilized to generate strong mathematical and analytical tools.

This publication is a wonderful source for college students and pros on top of things thought, robotics, engineering, special effects, econometrics, and any region that calls for the development of curves in accordance with units of uncooked data.

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**Additional info for Control Theoretic Splines: Optimal Control, Statistics, and Path Planning (Princeton Series in Applied Mathematics)**

**Example text**

In a similar fashion, we will be able to solve more and more involved optimal control problems that arise in the control theory literature, as well as in statistics and numerical analysis. EditedFinal September 23, 2009 Chapter Three EIGHT FUNDAMENTAL PROBLEMS In this chapter, we introduce a series of eight fundamental problems in the areas of interpolation and smoothing of increasing complexity. These problems will serve as basic building blocks for the developments in later chapters; in particular, we show that, although these eight problems have their origins in optimal control theory, statistics, and numerical analysis, they can be addressed in a unified manner.

16) can be reformulated as x(T ) = eAT x0 + Λu. 2 Controllability and the Point-to-Point Transfer Problem The point-to-point transfer problem considers whether it is possible to drive x from x(0) = x0 to a given x(T ) = xT . 17), the point-topoint transfer problem is exactly that of determining if there exists a u such that xT − eAT x0 = Λu, or equivalently if xT − eAT x0 ∈ R(Λ), where the range space of Λ is given by R(Λ) = {z ∈ Rn | ∃u ∈ L2 such that z = Λu}. Since L2 is an infinite-dimensional vector space, it may not be a particularly easy task to characterize this range space.

EditedFinal September 23, 2009 30 CHAPTER 3 This problem can be easily solved using the techniques discussed in the previous chapter. In fact, the affine variety in L2 [0, T ] that we are interested in is defined through the constraints Vα = {u ∈ L2 | αi = Lti (u), i = 1, . . , N }. Following the procedure from the projection theorem, we first have to construct the orthogonal complement to the linear subspace defined by V0 = {u ∈ L2 [0, T ] | Lti (u) = 0, i = 1, . . , N }. It is straightforward to see that this set is the same as the set spanned by the functions ti (s).