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Generic properties of singular trajectories. (English) Zbl 0907.93020
The authors consider the singular extremals of a control system on a $$\sigma$$-compact $$C^\infty$$ manifold $$M$$. The system is of type $\dot x(t)=F_0(x(t))+u(t)F_1(x(t)),\quad t\in J\equiv [T_1,T_2].$ Let $$H_0$$, $$H_1$$ be the Hamiltonians canonically defined by $$F_0$$, $$F_1$$ on $$T^*M$$, and let $$\vec{H_0}$$, $$\vec{H_1}$$ be the associated vector fields. The singular extremals are the solutions of the system on $$T^*M$$ $\dot z(t)=\vec{H_0}(z(t))+u(t)\vec{H_1}(z(t))$ satisfying the Pontryagin Maximum Principle. A singular extremal is said to be of minimum order if the set $\{t\in J :\{\{H_0,H_1\},H_1\}(z(t))\neq 0\}$ is dense in $$J.$$ The main result proved states that there is an open dense subset $$G$$ in the space of couples of $$C^\infty$$ vector fields with the $$C^\infty$$ Whitney topology such that the control system associated to any couple $$(F_0,F_1)\in G$$ has only minimal order singular trajectories, moreover each extremal trajectory is determined, up to a constant, by its projection on $$M$$.

##### MSC:
 93B29 Differential-geometric methods in systems theory (MSC2000) 49N60 Regularity of solutions in optimal control
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##### References:
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