Sufficient condition for optimal control with functional having second order state derivatives

Question

I have been searching for a Theorem which gives sufficient conditions for optimal control problem with state constraints in which the functional has second order state derivates, i.e., the cost function is of the form $$ J = \int_0^1 F(x(t),\dot x(t),\ddot x(t),u(t),t) dt.$$ The problem which I am dealing with has cost function $$ J = \int_0^1 (\ddot x^2 +\ddot y^2) dt ,$$ system dynamics $$\dot x=vcos(\theta), $$$$\dot y=vsin(\theta), $$ where $(x,y)$ is position of the object, $v,\theta$ are the two controls where $v$ is constant speed and $\theta(t)$ is the variable angle(direction) and constraints of the form $$ (x(t)-0.4)^2+(y(t)-0.5)^2 \geq 0.1, $$ $$ (x(t)-0.8)^2+(y(t)-1.5)^2 \geq 0.1. $$ The object is starting at the origin and final point is say $(1.2,1.6)$ at $t=1$. Can anyone please point me to any material related to this ?

Answer 1

If you compute the time derivative of your state equations and use the fact that $v$ is constant, then you get

$$\ddot{x}=-v\dot{\theta}\sin(\theta),\ \ddot{y}=v\dot{\theta}\cos(\theta).$$

Your cost becomes

$$J=\int_0^1\bar{u}(t)^2v^2dt$$

with the state equations

$$\dot{x}=v\cos(\theta),\ \dot{y}=v\sin(\theta),\ \dot{\theta}=\bar u.$$

Then, it is a standard optimal control problem that can be resolved using Pontryagin's Maximum Principle. This can be found on any textbook on optimal control. Note, however, that your problem may not admit an optimal control closed-form expression.