On bang-bang control policies
In this paper it is proposed a method for the determination of the optimal distribution of N switching points for a bang-bang control applied to a differential system.
After pointing the necessary conditions to be verified by such swit-ching points it is showed the existence of an optimal policy for a fixed number N of them.
Once characterized these points thru the application of the Pontryagin principle the problem, considered till now in the space of step functions, is put into the L1 space, in order to show the existence of a minimizing succession of the am plified problem and analized its correspondence to an optimal policy.
After reducing the problem into one of optimization on a convex K of IRn there are added considerations which let us, with the proposed method, obtain the optimal also with a number of switching points n less than the predetermined N.
Now it is proved that the function to optimize is of C2 class in K and the applied methods are these of the projected gradient and the conjugated gradient conveniently penalized.
Finally, the obtained algorithms are applied in one example: the shutdown policy of a nuclear reactor where the optimun is obtained with a finite number of switchings; this number remains constant although increasing values of N are proposed.
KeywordsOptimal Policy Conjugate Gradient Method Switching Point Projected Gradient Method Bang Control
- 1.E. POLAK. "Computational methods in optimization". Academic Press 1971.Google Scholar
- 2.E.B. LEE, L. MARKUS. "Foundations of Optimal Control Theory". Wiley. 1968.Google Scholar
- 3.L.S. PONTRYAGIN, V.G. BOLTYANSKII, R.V. GAMKERELIDZE, E.F. MISCHENKO. "Mathematical Theory of Optimal Processus". New York, Wiley 1962.-Google Scholar