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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 \(M\).

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