Projected dynamical systems have evolved out of the desire to dynamically model the behaviour of nonstatic solutions in equilibrium problems over some parameter, typically take to be time. This dynamics differs from that of ordinary differential equations in that solutions are still restricted to whatever constraint set the underlying equilibrium problem was working on, e.g. nonnegativity of investments in financial modeling, convexpolyhedral sets in operations research, etc. One particularly important class of equilibrium problems which has aided in the rise of projected dynamical systems has been that of variational inequalities. The formalization of projected dynamical systems began in the 1990s. However, similar concepts can be found in the mathematical literature which predate this, especially in connection with variational inequalities and differential inclusions.
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 spaceX. The normal cone to the setK at the point x in K is given by The tangent cone to the set K at the point x is given by The projection operator of a point x in X to K is given by the point in K such that for every y in K. The vector projection operator of a vector v in X at a point x in K is given by
Given a closed, convex subset K of a Hilbert space X and a vector field -F which takes elements from K into X, the projected differential equation associated with K and -F is defined to be On the interior of K solutions behave as they would if the system were an unconstrained ordinary differential equation. However, since the vector field is discontinuous along the boundary of the set, projected differential equations belong to the class of discontinuous ordinary differential equations. While this makes much of ordinary differential equation theory inapplicable, it is known that when -F is a Lipschitz continuous vector field, a unique absolutely continuous solution exists through each initial pointx=x0 in K on the interval. This differential equation can be alternately characterized by or The convention of denoting the vector field -F with a negative sign arises from a particular connection projected dynamical systems shares with variational inequalities. The convention in the literature is to refer to the vector field as positive in the variational inequality, and negative in the corresponding projected dynamical system.
