Projected Dynamical System - Projections and Cones

Projections and Cones

Any solution to our projected differential equation must remain inside of our constraint set K for all time. This desired result is achieved through the use of projection operators and two particular important classes of convex cones. Here we take K to be a closed, convex subset of some Hilbert space X.

The normal cone to the set K at the point x in K is given by

$N_K(x) = \{ p \in V | \langle p, x - x^* \rangle \geq 0, \forall x^* \in K \}.$

The tangent cone (or contingent cone) to the set K at the point x is given by

$T_K(x) = \overline{\bigcup_{h>0} \frac{1}{h} (K-x)}.$

The projection operator (or closest element mapping) of a point x in X to K is given by the point in K such that

$\| x-P_K(x) \| \leq \| x-y \|$

for every y in K.

The vector projection operator of a vector v in X at a point x in K is given by

$\Pi_K(x,v)=\lim_{\delta \to 0^+} \frac{P_K(x+\delta v)-x}{\delta}.$

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