Projected Dynamical System - Projected Differential Equations

Projected Differential Equations

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

\frac{dx(t)}{dt} = \Pi_K(x(t),-F(x(t))).

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 point x(0)=x0 in K on the interval .

This differential equation can be alternately characterized by

\frac{dx(t)}{dt} = P_{T_K(x(t))}(-F(x(t)))

or

\frac{dx(t)}{dt} = -F(x(t))-P_{N_K(x(t))}(-F(x(t))).

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.

Read more about this topic:  Projected Dynamical System

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