Projected dynamical system: Difference between revisions

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'''Projected dynamical systems''' is a [[mathematics|mathematical]] theory investigating the behaviour of [[dynamical system]]s where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of [[Optimization (mathematics)|optimization]] and [[Equilibrium point|equilibrium]] problems and the dynamical world of [[ordinary differential equations]]. A '''projected dynamical system''' is given by the [[flow (mathematics)|flow]] to the '''projected differential equation'''

:<math>

\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>

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

</math>

Which is just the Gateaux Derivative computed in the direction of the Vector field

== Projected Differential Equations ==

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</math>

On the [[Interior (topology)|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]] solutionssolution existexists through each initial point ''x(0)=x₀'' in ''K'' on the interval <math>[0,\infty)</math>.

This differential equation can be alternately characterized by

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== See also ==

* [[Differential variational inequality]]

* [[Dynamical systems theory]]

* [[Ordinary differential equation]]

* [[Variational inequality]]

* [[Differential variational inequalityinclusion]]

* [[Complementarity theory]]

== 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).

* Brogliato, B., and Daniilidis, A., and [[Claude Lemaréchal|Lemaréchal, C.]], and Acary, V., "On the equivalence between complementarity systems, projected systems and differential inclusions", ''Systems and Control Letters'', vol.55, pp.45-51 (2006)

[[Category:Differential equations]]

[[Category:Dynamical systems]]