# zbMATH — the first resource for mathematics

The generic local structure of time-optimal synthesis with a target of codimension one in dimension greater than two. (English) Zbl 0951.49027
Summary: In previous papers we gave in dimension 2 and 3 a classification of generic synthesis of analytic systems $$\dot v(t)= X(v(t)) +u(t)Y(v(t))$$ with a terminal submanifold $$N$$ of codimension one when the trajectories are not tangent to $$N$$. We complete here this classification in all generic cases in dimension 3, giving a topological classification and a model in each case. We prove also that in dimension $$n\geq 3$$, out of a subvariety of $$N$$ of codimension three, we have described all the local synthesis.

##### MSC:
 49K15 Optimality conditions for problems involving ordinary differential equations 34H05 Control problems involving ordinary differential equations 93C15 Control/observation systems governed by ordinary differential equations
Full Text:
##### References:
 [1] A. A. Agrachev, Exponential mappings for contact sub-Riemannian structures.J. Dynam. Control Syst. 2 (1996) 321–358. · Zbl 0941.53022 [2] R. Benedetti and J.-J. Risler, Real algebraic and semialgebraic sets.Hermann, Paris, 1990. · Zbl 0694.14006 [3] B. Bonnard, Rapport préliminaire sur la conduite optimale des réacteurs chimiques de type batch.Prépubl. Univ. de Bourgogne, 1995. [4] B. Bonnard, M. Chyba, and H. Heutte, Contrôle optimal géométrique appliqué.Prépubl. Univ. de Bourgogne, 1995. [5] B. Bonnard and I. Kupka, Théorie des singularités de l’application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal.Forum Math. 5 (1993), 111–159. · Zbl 0779.49025 [6] B. Bonnard and I. Kupka, Generic properties of singular trajectories.Ann. Inst. H. Poincaré. (to appear). · Zbl 0907.93020 [7] B. Bonnard, G. Launay, and M. Pelletier, Classification générique de synthèses temps minimales avec cible de codimension un et applications.Ann. Inst. H. Poincaré. (to appear). · Zbl 0891.49012 [8] B. Bonnard, and M. Pelletier, Time minimal synthesis for planar systems in the neighborhood of a terminal manifold of codimension one. Summary.J. Math. Syst. Estimat. and Control 5 (1995), No. 3, 379–381. · Zbl 0852.49014 [9] I. Ekeland, Discontinuités des champs hamiltoniens et existence de solutions optimales en calcul des variations.Publ. IHES (1977), No. 47, 1–32. · Zbl 0447.49015 [10] M. Golubitsky and V. Guillemin, Stable mappings and their singularities.Springer-Verlag, New-York, 1973. · Zbl 0294.58004 [11] H. Hermes, Lie algebra of vector fields and local approximation of attainable sets.SIAM J. Control and Optimiz. 26 (1978), No. 5, 715–727. · Zbl 0388.49025 [12] I. Kupka, Geometric theory of extremals in optimal control problems, I. The fold and Maxwell cases.Trans. Am. Math. Soc. 299 (1973), 225–243. · Zbl 0606.49016 [13] E. B. Lee and L. Markus, Foundations of otimal control theory.Wiley, New-York, 1977. [14] L. Pontriaguine et al., Théorie mathématique des processus optimaux.Mir, Moscow, 1974. [15] H. Schättler, The local structure of time optimal trajectories under generic conditions.SIAM J. Control and Optimiz. 26 (1988), 899–918. · Zbl 0656.49007 [16] H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: TheC nonsingular case.SIAM J. Control and Optimiz. 25 (1987), 433–465.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.