In this case, instead of the boundary condition V(Tf) =0 we have V(Tf,xf)=0 which is not enough to solve the related PDE.
I wonder if it is possible to extend/modify the classical formulation to come up with a well posed problem. Obviously, we assume that the system is controllable and it is always possible to drive the system to the desired end-point xt.
Hello.
Fixed-endpoint problems are typical in optimal control. Look, for instance, at:
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1649860&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D1649860
Hi Pavel,
thank you for the link. This particular approach is pretty interesting, but there is one fundamintal difficulty. All methods that are based on some manipulations with Hamiltonian equations in (x,p) implicitely imply that the control lies within the boundaries. In this way we drasticaly reduce the set of control problems under consideration as the key feature of optimal control is that the control signal may lie on the border.
In the paper on the origin of Pontryagin's maximum principle by Gamkrelidze (if I'm not mistaking) the auther mentioned that this particular aspect, i.e. the boundedness of controls, led Pontryagin to the idea that we have to move from the classical variational calculus to a novel theory which later culminated in the celebrated PMP.
Hello,
I hope that I understood the question well. Fixed final state-fixed final time is quite common situation. For examples, dynamic programming has been applied extensively to calculate the optimal power management strategy for hybrid electric vehilces, where the final time is fixed to the duration of the driving cycle and the target final state-of-charge of the battery is also fixed. In this case, the cost-to-go at (tf,xf) is set to zero, and it is set to a large value for all other final states.
PMP has been also used in hybrid vehicle, however, there is still no well-established method to deal with the state boundaries (upper and lower bounds); however, there exist some heuristic methods that work quite well and give very good agreement with the results of dynamic programming.
Hi Farouk,
the approach you have described is based on the penalization of the deviations from the desired end-point. However, in my view, this does not guarantee that you arrive *exactly* at that point except for the case that you have infinitely big penalty for any x =/= xf.
I wonder if there is some formal approach to this problem...
You have a equality constraint for the optimal control problem: V(t_f,x_f)=0. Just add a constant multiplier (if V is a scalar function) to the integrated Lagrangian and obtain the equations for the solution. That constant multiplier will have an equation.
Dear Zigang Pan,
could you be little bit more specific? I'm not sure if I got the idea...
My understanding is that you want to terminate the optimization when V(t_f,x_f)=0, where t_f and x_f are not determined. Then, it is just an equality constraint on the optimization problem that admits a constant Lagrange multiplier (since the constraint is not pointwise in time for the entire duration of the optimization). It is easy to deal with equality constraint according to usual method. (One can't add the multiplier term in the Hamiltonian since that corresponds to a constraint for the entire duration of the optimization. ) I hope that I made myself clear now. I need to check books if you want me to give you equations, and I am lazy to do that.
In dynamical programming, V(t,x) is the optimal value of the cost function that we get when starting at time t from point x. And, respectively, V(tf,xf)=0 is not a termination condition, but the boundary condition for the Hamilton-Jacobi-Bellman partial-differential equation which I have to solve in order to get V. This condition says that when starting at final time=tf from the final point=xf, the best reward we can get is equal to 0.
Thus, the described approach seems to be not very relevant to the problem described.
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