مرکز منطقه ای اطلاع رساني علوم و فناوري - Sufficient conditions for controllability

Abstract :

The problem is to find sufficient conditions for the system $\\dot{x}(t)= f(x((t))+ \\sum _{i = 1}^{m} u_{i}(t)g_{i}(x(t)), x(0)=x_{0} \\in M$ to be controllable. Here $M$ is a connected ${cal C}^{\\infty }$ $n$ -dimensional manifold, $f$ , $g_{1}, \\cdots ,g_{m}$ are complete ${cal C}^{\\infty }$ vector fields on $M$ , and $u_{1}, \\cdots , u_{m}$ are real-valued controls. If $m = n - 1, M,f, g_{1}, \\cdots ,g_{n-1}$ are real-analytic, $M$ is simply connected, and $g_{1}, \\cdots ,g_{n-1}$ are linearly independent on $M$ , then necessary and sufficient conditions are known. For the case of our ${cal C}^{\\infty }$ system with general $m$ , we assume that the space spanned by the Lie algebra $L_{A}$ generated by $f, g_{1}, \\cdots ,g_{m}$ and successive Lie brackets has constant dimension $p$ on $M$ and the algebra $L_{A}^{\\prime }$ generated by $g_{1}, \\cdots ,g_{m}$ and successive Lie brackets has constant dimension $p^{\\prime } \\leq p$ on $M$ . If $p^{\\prime }=p$ , Chow\´s Theorem implies controllability for a $p$ -dimensional submanifold of $M$ containing $x_{0}$ . If $p^{\\prime }< p$ , sufficient conditions are found involving the computation of certain Lie brackets at points where the vector field $f$ is tangent to the integral manifolds of $L_{A}^{\\prime }$ . Here we assume that every integral manifold of $L_{A}^{\\prime }$ contains such a point.