David H. Jacobson

New necessary conditions of optimality for control problems with state-variable inequality constraints☆☆☆

Abstract Necessary conditions of optimality for state-variable inequality constrained problems are derived which differ from those of Bryson, Denham, and Speyer with regard to the behavior of the adjoint variables at junctions of interior and boundary arcs. In particular, it is shown that the earlier conditions under-specify the behavior of the adjoint variables at the junctions. An example is used to demonstrate that the earlier conditions may yield non-stationary trajectories. For a certain class of problems, it is shown that only boundary points, as opposed to boundary arcs, are possible. An analytic example illustrates this behavior.

Necessary and sufficient conditions for optimality for singular control problems: A limit approach

Abstract Necessary and sufficient conditions for optimality for singular control problems are presented for the case where the extremal path is totally singular. The singular second variation is converted into a nonsingular one by addition of a quadratic functional of the control; a parameter 1 ϵ multiplies this added functional. By allowing ϵ to approach infinity the optimality conditions are deduced for the singular problem from the limiting optimality conditions of the synthesized nonsingular second variation. The resulting conditions are Jacobsons sufficient conditions in slightly modified form. In a companion paper necessity of Jacobsons conditions for a class of singular problems is demonstrated by exploiting the Kelley transformation technique which converts the singular second variation into a nonsingular one in a reduced dimensional state space.

An Algorithm that Minimizes Homogeneous Functions of N Variables in N + 2 Iterations and Rapidly Minimizes General Functions

Abstract A new algorithm for function minimization is presented. The new algorithm is based upon homogeneous functions rather than quadratic models. A consequence of this is that (n + 2) step convergence is obtained for homogeneous functions and that no one-dimensional search is required. Preliminary numerical tests indicate that on general functions the algorithm is superior to the well-known Fletcher and Powell method.

A general sufficiency theorem for the second variation

Abstract A general sufficiency theorem is presented for a class of quadratic minimization problems. This allows one to treat both singular and nonsingular quadratic functionals (second variations) as special cases of a general “partially singular” functional.

On singular arcs and surfaces in a class of quadratic minimization problems

Abstract A class of quadratic minimization problems is studied whose optimal control functions are partially singular. An explicit expression is obtained for the (n − 1)-dimensional singular surface and it is shown that the optimal value function is twice continuously differentiable across this surface; this allows the piecing together of known sufficiency conditions for totally singular and totally bang-bang arcs to obtain sufficient conditions for optimality for this class of partially singular problems. A neighboring optimal feedback control law is suggested by these results. The closing sections of the paper are concerned with sufficient conditions for nonexistence of optimal singular controls in quadratic minimization problems.