Time optimality of non-smooth singular trajectories of an affine system with scalar input in \(R^3\) (Q2771544)

The author considers the non-smooth singular trajectories of the control system \(\dot q(t)=X(q(t))+u(t)Y(q(t))\) with a scalar control \(u\), where \(X,Y\) are given germs of \(C^{\infty}\) vector fields. Let \(q(t)\in U\), where \(U\) is an open neighborhood of a fixed point \(q_0\in R^3\), and let the following conditions for the Lie brackets of the considered system hold true at the point \(q_0\): (C1) \(D''(q_0)=\det(X,Y,[X,Y])\neq 0\); (C2) \(D(q_0)=\det(Y,[X,Y],\text{ad}^2Y\cdot X)=0\); (C3) \(D'(q_0)=\det(Y,[X,Y],\text{ad}^2X\cdot Y)=0\); (C4) \(\Delta=\det(Y,[X,Y],\text{ad}^3Y\cdot X)\neq 0\); (C5) \(\Delta'=\det(Y,[X,Y],\text{ad}^3X\cdot Y)\neq 0\); (C6) Conditions \(D=0\) and \(D'=0\) determine the transversal surfaces in \(U\), and vector field \(Y\) is transversal to these surfaces in \(U\); (C7) \(\Delta'\Delta>0\).NEWLINENEWLINENEWLINEThe following results are presented. Under conditions (C1)--(C7) non-smooth singular extremal trajectory looses its strong time optimality at the point \(q_0\). Let \(u:[0,T]\to[-C,C]\). Under conditions (C1)-(C7) there exists \(C_0\) such that: if \(C>C_0\), then the corresponding singular trajectory \(q_{s}(\cdot)\) is not time optimal, but if \(C<C_0\), then this trajectory is time optimal. The weak optimality is considered, too.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00042].