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\frac{dx(t)}{dt} = \Pi_K(x(t),-F(x(t)))
</math>
where ''K'' is our constraint set. Differential equations of this form are notable for having a discontinuous vector field.
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== History of projected dynamical systems ==
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 [[finance|financial]] modeling, [[convex set|convex]] [[Polyhedron|polyhedral]] 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 inequality|variational inequalities]].
The formalization of projected dynamical systems began in the 1990s in Section 5.3 of the paper of Dupuis and Ishii. 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 ==
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</math>
The ''[[tangent cone]]'' (or ''contingent cone'') to the set ''K'' at the point ''x'' is given by
:<math>
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</math>
Which is just the Gateaux Derivative computed in the direction of the Vector field
== Projected Differential Equations ==
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== References ==
* Henry, C., "Differential equations with discontinuous right-hand side for planning procedures", ''J. Econom. Theory'', 4:545-551, 1972.
* Henry C., "An existence theorem for a class of differential equations with multivalued right-hand side", ''J. Math. Anal. Appl.'', 41:179-186, 1973.
* Aubin, J.P. and Cellina, A., ''Differential Inclusions'', Springer-Verlag, Berlin (1984).
* Dupuis, P. and Ishii, H., ''On Lipschitz continuity of the solution mapping to the Skorokhod Problem, with applications'', Stochastics and Stochastics Reports, 35, 31-62 (1991).
* Nagurney, A. and Zhang, D., ''Projected Dynamical Systems and Variational Inequalities with Applications'', Kluwer Academic Publishers (1996).
* Cojocaru, M., and Jonker L., ''Existence of solutions to projected differential equations on Hilbert spaces'', Proc. Amer. Math. Soc., 132(1), 183-193 (2004).
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[[Category:Differential equations]]
[[Category:Dynamical systems]]
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