The second problem from (8.20) is to determine an expression for
. This is where the laws of physics, such as the acceleration of rigid bodies due to applied forces and gravity. The most common case is time-invariant dynamical systems, in which depends only on the current state and not the particular time. This means each component is expressed as
for some given vector-valued function
. This can be written in compressed form by using and to represent -dimensional vectors:
The expression above is often called the state transition equation because it indicates the state's rate of change.
Here is a simple, one-dimensional example of a state transition equation:
This is called a linear differential equation. The velocity roughly doubles with the value of . Fortunately, linear problems can be fully solved ``on paper''. The solution to (8.27) is of the general form
in which is a constant that depends on the given value for .
Steven M LaValle