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Normality and Nondegeneracy of the Maximum Principle in Optimal Impulsive Control Under State Constraints

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Abstract

We investigate nondegenerate and normal forms of the maximum principle for general, free end-time, impulsive optimal control problems with state and endpoint constraints. We introduce constraint qualifications sufficient to avoid degeneracy or abnormality phenomena, which do not require any convexity and impose the existence of an inward pointing velocity just on the subset of times, in which the extended optimal trajectory has an outward pointing velocity (w.r.t. the state constraint). These conditions extend to impulsive problems some conditions, recently proposed by F. Fontes and H. Frankowska, for conventional optimization problems. The nontriviality of this extension is illustrated through some examples.

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Notes

  1. The space-time system (7) is rate-independent, hence our using only controls such that \(w_0+\left| w\right| =1\) is an useful but arbitrary choice (see, e.g., [5] for more details).

  2. Since every \(L^1\)-equivalence class contains Borel measurable representatives, we are tacitly assuming that all \(L^1\)-maps we are considering are Borel measurable.

  3. Let us recall that (2) is called commutative if the Lie brackets \([g_i,g_j]\equiv 0\) for all i, j.

  4. We use \(|\varvec{\eta }_i|\) to denote the total variation function of \(\varvec{\eta }_i\).

  5. For any \(r\in \mathbb {R}\), \(\delta _{\{r\}}\) is the Dirac unit measure concentrated at r.

  6. For any \(w\in \mathbb {R}^m\), when \(w=0\) we mean that \(\frac{w}{|w|}=0\).

  7. A cone \({\mathcal K}\subseteq \mathbb {R}^k\) is pointed if it contains no line, i.e., if z, \(-z\in {\mathcal K}\) implies that \(z=0\).

  8. Since the scalar product is bilinear, \(N_\varOmega ({{x}^0})\) can be replaced by the Clarke normal cone, as in [15].

  9. Given an interval \(I\subset \mathbb {R}\), we denote by \(\chi _{_{I}}\) the characteristic function of I.

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Acknowledgements

This research is partially supported by the Padua University Grant SID 2018 “Controllability, stabilizability and infimun gaps for control systems”, prot. BIRD 187147, and by the Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), Italy. The authors are very grateful to the anonymous referees for their constructive criticism to improve the article and for the helpful bibliographic suggestions.

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  1. Department of Mathematics “Tullio Levi-Civita”, University of Padua, Padua, Italy

    Monica Motta & Caterina Sartori

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  1. Monica Motta
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  2. Caterina Sartori
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Correspondence to Monica Motta.

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Communicated by Roland Herzog.

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Motta, M., Sartori, C. Normality and Nondegeneracy of the Maximum Principle in Optimal Impulsive Control Under State Constraints. J Optim Theory Appl 185, 44–71 (2020). https://doi.org/10.1007/s10957-020-01641-w

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