Abstract
In this concluding chapter, an extension of the classical control problem is given in the most general nonlinear case. The essential matter is that now the control variable is not split into conventional and impulsive types, while the dependence on this unified control variable is not necessarily affine. By combining the two approaches, the one based on the Lebesgue discontinuous time variable change, and the other based on the convexification of the problem by virtue of the generalized controls proposed by Gamkrelidze, a fairly general extension of the optimal control problem is constructed founded on the concept of generalized impulsive control. A generalized Filippov-like existence theorem for a solution is proved. The Pontryagin maximum principle for the generalized impulsive control problem with state constraints is presented. Within the framework of the proposed approach, a number of classic examples of essentially nonlinear problems of calculus of variations which allow for discontinuous optimal arcs are also examined. The chapter ends with seven exercises.
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Notes
- 1.
A small historical note: The equation of the catenary curve was derived by Leibniz, Huygens, and Johann Bernoulli in 1691. They were the first to discover that this curve is a hyperbolic cosine, but not a parabola, as had been thought before. The shape of the soap bubble in (7.1) is, as we see from (7.3), also formed by the catenary. (This was noticed by Euler.)
- 2.
It is also possible to consider the case of finite limit. But then, as it will become clear from the forthcoming exposition, the trajectory discontinuities will not arise in the extension. Therefore, this case is not of interest for this book. (It is already encompassed in the known theory.)
- 3.
Note that the solution to the extended problem in this case may happen to lie in \(\mathrm{{L}}_1\). Then, the transition to discontinuous trajectories is already redundant.
- 4.
Note that, these are two different types of convergence; see [6].
- 5.
This lemma, in particular, says that any generalized control can be weakly approximated by conventional controls, that is, in the sense of weak convergence of generalized controls.
References
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Arutyunov, A., Karamzin, D., Lobo Pereira, F. (2019). General Nonlinear Impulsive Control Problems. In: Optimal Impulsive Control. Lecture Notes in Control and Information Sciences, vol 477. Springer, Cham. https://doi.org/10.1007/978-3-030-02260-0_7
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DOI: https://doi.org/10.1007/978-3-030-02260-0_7
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