Necessary optimality conditions for a singular surface in the form of synthesis

Abstract

For a linear control problem using the traditional open-loop approach, a new representation for the singular control and generalized, invariant conditions for optimality are found. The phase portrait of a nonlinear control problem is considered in the neighborhood of singular trajectories. The singular paths form a hypersurface, approached by regular paths from both sides. The Bellman function for this problem is a classical (smooth) solution to a first-order PDE with nonsmooth Hamiltonian over two smooth (regular) branches, related to the halfneighborhoods of the surface. These solutions are at least twice differentiable and have first discontinuous derivatives of odd order. The invariant form for these necessary conditions is found in terms of Jacobi (Poisson) brackets, consisting of several equalities and inequalities. The latter relations guarantee the validity of the Kelley condition as well as the geometrical constraints for the singular control variables. Thus, the Kelley condition appears to be just a certain property of a smooth solution to a first-order PDE with nonsmooth Hamiltonian. All the relations, including the Hamiltonian equations of singular motion, do not use singular controls; they are based on regular Hamiltonians depending only upon the state vector and the gradient of the Bellman function (adjoint vector).