Difference equation in the context of "State (controls)"

In mathematics and particularly in dynamical systems, an initial condition is the initial value (often at time ${\displaystyle t=0}$) of a differential equation, difference equation, or other "time"-dependent equation which evolves in time. The most fundamental case, an ordinary differential equation of order k (the number of derivatives in the equation), generally requires k initial conditions to trace the equation's evolution through time. In other contexts, the term may refer to an initial value of a recurrence relation, discrete dynamical system, hyperbolic partial differential equation, or even a seed value of a pseudorandom number generator, at "time zero", enough such that the overall system can be evolved in "time", which may be discrete or continuous. The problem of determining a system's evolution from initial conditions is referred to as an initial value problem.

A finite difference is a mathematical expression of the form f(x + b) − f(x + a). Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.

The difference operator, commonly denoted ${\displaystyle \Delta }$, is the operator that maps a function f to the function ${\displaystyle \Delta [f]}$ defined by$${\displaystyle \Delta [f](x)=f(x+1)-f(x).}$$A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.