Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations.
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 financial modeling, convex 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 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 re ult 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 space X.
Ordinary differential equations arise in many different contexts throughout mathematics and science (social and natural) one way or another, because when describing changes mathematically, the most accurate way uses differentials and derivatives (related, though not quite the same). Since various differentials, derivatives, and functions become inevitably related to each other via equations, a differential equation is the result, governing dynamical phenomena, evolution and variation. Often, quantities are defined as the rate of change of other quantities (time derivatives), or gradients of quantities, which is how they enter differential equations.
Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), geology (weather modelling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.