Sufficient conditions for a path-connected set of local solutions to an optimal control problem (Q2810046)

The problems are of the form NEWLINE\[NEWLINE\begin{aligned} \underset{q, u}{\text{minimize}} \ \ C(q, u) &= \int_0^T g(q(t), u(t)) dt \\ NEWLINE\text{subject to} \ \ \quad q'(t) &= f(q(t), u(t)) , \\ NEWLINEq(0) &= 0 , \ \ q(T) = b . \end{aligned} \tag{1} NEWLINE\]NEWLINEThe problem under study is whether the optimal trajectory of (1) with final condition \(q(T) = 1\) can be continuously deformed into the optimal trajectory with final condition \(q(T) = -1\) in such a way that each trajectory along the deformation is a local solution of the optimal control problem for some choice of \(q (T).\) Systems having this property are called \textit{path connected}. The authors derive necessary and sufficient conditions for optimality (the first based on Pontryagin's maximum principle) and show that if the Hamiltonian of the system satisfies a scale invariance property then the system is path connected. They also look at the effect of additional constraints on path connectedness. NEWLINENEWLINEThe main motivation is the control system where the ends of a thin rod are held by robotic grippers. The problem is how to move the gripper holding the right end in such a way that the rod moves from an initial shape to a target shape.