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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.)

Roland
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1 Answers1

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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?

Rafael Rojas
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