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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'''
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| :<math>
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| \frac{dx(t)}{dt} = \Pi_K(x(t),-F(x(t)))
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| </math>
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| 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 ==
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| 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]].
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| 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.
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| == Projections and Cones ==
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| 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 cone]]s. Here we take ''K'' to be a [[closed set|closed]], [[convex set|convex]] subset of some [[Hilbert space]] ''X''.
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| The ''normal cone'' to the set ''K'' at the point ''x'' in ''K'' is given by
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| :<math>
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| N_K(x) = \{ p \in V | \langle p, x - x^* \rangle \geq 0, \forall x^* \in K \}.
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| </math>
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| The ''[[tangent cone]]'' (or ''contingent cone'') to the set ''K'' at the point ''x'' is given by
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| :<math>
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| T_K(x) = \overline{\bigcup_{h>0} \frac{1}{h} (K-x)}.
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| </math> | |
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| The ''projection operator'' (or ''closest element mapping'') of a point ''x'' in ''X'' to ''K'' is given by the point <math>P_K(x)</math> in ''K'' such that
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| :<math> | |
| \| x-P_K(x) \| \leq \| x-y \|
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| </math>
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| for every ''y'' in ''K''.
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| The ''vector projection operator'' of a vector ''v'' in ''X'' at a point ''x'' in ''K'' is given by
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| :<math> | |
| \Pi_K(x,v)=\lim_{\delta \to 0^+} \frac{P_K(x+\delta v)-x}{\delta}.
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| </math>
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| == Projected Differential Equations ==
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| 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
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| :<math>
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| \frac{dx(t)}{dt} = \Pi_K(x(t),-F(x(t))).
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| </math>
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| 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]] solution exists through each initial point ''x(0)=x<sub>0</sub>'' in ''K'' on the interval <math>[0,\infty)</math>.
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| This differential equation can be alternately characterized by
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| :<math>
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| \frac{dx(t)}{dt} = P_{T_K(x(t))}(-F(x(t)))
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| </math>
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| or
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| :<math>
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| \frac{dx(t)}{dt} = -F(x(t))-P_{N_K(x(t))}(-F(x(t))).
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| </math>
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| 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.
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| == See also ==
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| * [[Differential variational inequality]]
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| * [[Dynamical systems theory]]
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| * [[Ordinary differential equation]]
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| * [[Variational inequality]]
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| * [[Differential inclusion]]
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| * [[Complementarity theory]]
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| == References ==
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| * Aubin, J.P. and Cellina, A., ''Differential Inclusions'', Springer-Verlag, Berlin (1984).
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| * Nagurney, A. and Zhang, D., ''Projected Dynamical Systems and Variational Inequalities with Applications'', Kluwer Academic Publishers (1996).
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| * 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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| * 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)
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| [[Category:Differential equations]]
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| [[Category:Dynamical systems]]
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Irwin Butts is what my wife enjoys to call me though I don't truly like being known as like that. Puerto Rico is where he and his spouse reside. To collect badges is what her family and her appreciate. Supervising is my profession.
Here is my weblog: over the counter std test