Extension of Pontryagin's principle

Question

Is anyone aware of an extension of Pontryagin's principle where the cost (more precisely, the Langrangian) may depend on the derivative of the control? So, instead of $\int_{t_0}^{t_1} L(t,x,u) dt + K(x(t_1))$, the cost would be given by $\int_{t_0}^{t_1} L(t,x,u,u') dt + K(x(t_1))$, where $t \mapsto x(t)$ denotes the state variables induced by the control $t \mapsto u(t)$. (I know that in many situations the optimal control function will be not differentiable, but let us assume that we are in a situation where this holds true.)

Answer 1

I have dealt with your same problem. Having a running cost term (the Lagrangian) that depends on the control is outside the statements of the Pontryagin Maximum Principle, Dynamic Programing and Krotov's extension principles. This is because choosing a control $u_i(t)$ means also choosing its derivative. Control (programs) are functions (of time), so each function comes with all its unique derivatives. At this point there is no difference between choosing $u_i$ to steer a system or choosing $\dot u_i$. It is absolutely valid to interpret $\dot u_i$ as a new control $v_i$ and $u_i$ as a new state variables $x_{n+1} = u_i$ s.t. $\dot x_{n+1} = v_i$. Here the general trick to solve your problem. Consider $$ I = F(T,x(T)) + \int_0^T L(t,x,u,\dot u)\ dt $$ subjec to $$ \dot x = f(t,x,u) $$ Now yow have to add a new set of variables $y$ and a new set of controls $v$ such that $$ \dot y = v $$ where $v$ coincides with the value of the derivatives of the controls $\dot u$. Now your problem will read $$ I = F(T,x(T)) + \int_0^T L(t,x,y,v) \ dr $$ subject to $$ \dot x = f(t,x,y) $$ and $$ \dot y = v $$

I found the same problem on the control of a wind turbine in order to maximize the energy subtracted from the wing current using the transmission ratio $\tau$ as a control variable . This can be done with actual technologies. In this case the equation of motions depends on the derivative of $\tau$. Could you tell me what systems are you dealing with?