In
an optimal control problem,one adjusts controls in a dynamic system to achieve
a goal. The underlying system can have a variety of types of equations such as
ordinary differential equations, partial differential equations, discrete
equations, stochastic differential equations or integro-difference equations.
We
are considering only systems of ordinary differential equations.
In
control of a single ordinary differential equation (ODE) we denote u (t) as the
control and x (t) as the state. The state function, x (t), satisfies the ODE
modeling scenario. The control affects the state ODE by
x’
(t)= g (t, x (t), u (t)),
Where x’
denote the derivative with respect to time t. Both u (t) and x (t)
affect the goal, which is called the objective functional. We seek to find an
optimal control and corresponding state that achieve the maximum (or minimum)
of our objective functional.
To
understand the basic idea of optimal control and the terminology, consider the
following example with very simple dynamics.
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