The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems
This article deals with the classical projection operator \(P_D\) of a Hilbert space \(H\) on a given closed convex set \(D \subset H\); the value of this operator at the element \(s \in D\) is defined as the nearest (to \(x\)) element \(P_Dx\) of \(D\). It is known that the operator \(P_D\) has directional derivative \(\Pi_D(x,v)\) for each \(x \in D\) and an arbitrary \(v \in H\). The authors prove the following statements: (I) \(\Pi_D(x,v) = v\) if \(x \in \text{int} \, D\); (II) \(\Pi_D(x,v) = v\) if \(x \in \partial D\) and \(\sup_{n \in S^* \cap N_D(x)} \;\langle v,n \rangle < 0\) only if \(\dim H < \infty\); if \(H\) is infinite-dimensional, this representation is not valid. (III) \(\Pi_D(x,v) = v - \langle v,n^* \rangle n^*\) if \(x \in \partial D\) and \(\sup_{n \in S^* \cap N_D(x)} \;\langle v,n \rangle > 0\), where \(n^*\) achieves the \(\sup_{n \in S^* \cap N_D(x)} \;\langle v,n \rangle\); moreover, \(n^*\) is unique. (IV) \(\Pi_D(x,v) = v\) if \(x \in \partial D\), \(\sup_{n \in S^* \cap N_D(x)} \;\langle v,n \rangle = 0\), and \(\sup_{n \in S^* \cap N_D(x)} \;\langle v,n \rangle\) has at least a maximizer \(n^0 \neq 0\). (V) \(\Pi_D(x,v) = v - \xi\), where \(\xi\) is an undetermined element in \(N_D(x)\), if \(x \in \partial D\) and we are in any case not considered in conclusions (I--IV); \(N_D(x) = \{\xi \in H| \;\langle \xi,y - x \rangle \;\leq 0, \;\text{for all} \;y \in D\}\). These statements generalize (to the infinite-dimensional case) results by \textit{P. Dupuis} [Stochastics 21, 63--96 (1987; Zbl 0614.60023); Ann. Oper. Res. 44, No.1--4, 9--42 (1993; Zbl 0785.93044)]. As applications, the problems of finding solutions \(x(\cdot): \;[0,\infty) \to H\) to the equations \[ \frac{dx(t)}{dt} = \Pi_D(x(t); - f(x,t)), \quad x(0) = x_0 \in D, \] \[ \frac{dx(t)}{dt} \in - f(x) - N_D(x), \quad x(0) = x_0 \in D, \] and \[ \left[\frac{dx(t)}{dt}\right]_w \in - f(x) - N_D(x), \quad x(0) = x_0 \in D \] (the left hand side of the latter equation is defined as an element \(z \in H\) satisfying the equation \(\lim_{\tau \to t} \;\frac{\langle \varphi,x(\tau) - x(t) \rangle}{\tau - t} = \langle \varphi,z \rangle\) for any \(\varphi \in H\)) are considered.
scientific article; zbMATH DE number 2111349