Homogeneous Tangent Vectors and High Order Necessary Conditions for Optimal Controls

Summary: The author introduces and analyzes "homogeneous" tangent vectors which provide high-order approximations to the attainable set for an affine control system of the form $dot x=X_0(x)+sum^m_{j=1}u_jX_j(x)$. Homogeneous tangent vectors are defined relative to one-parameter families of dilations $delta^r_epsiloncolonR^ntoR^n$ on $R^n$. Adjoint equations associated with the corresponding homogeneous variational equation are derived and used to transport homogeneous tangent vectors along the flow of a reference trajectory. These constructions are then used to derive a homogeneous high-order test for optimality of control problems in Mayer form without terminal constraints. Essentially, it is shown that if $v_nu(t)$ is a homogeneous tangent vector with respect to a dilation $delta^r_epsilon$ generated by a control variation, then it is a necessary condition for optimality that $p(t)v_nu(t)leq0$, where $p(t)$ denotes the solution of the corresponding homogeneous adjoint equation.