Future planetary explorations envisioned by the National Research Council's (NRC's) report titled Vision and Voyages for Planetary Science in the Decade 2013-2022, developed for NASA Science Mission Directorate (SMD) Planetary Science Division (PSD), seek to reach targets of broad scientific interest across the solar system. This goal requires new capabilities such as innovative interplanetary trajectories, precision landing, operation in close proximity to targets, precision pointing, multiple collaborating spacecraft, multiple target tours, and advanced robotic surface exploration. Advancements in Guidance, Navigation, and Control (GN&C) and Mission Design in the areas of software, algorithm development and sensors will be necessary to accomplish these future missions. This paper summarizes the key GN&C and mission design capabilities and technologies needed for future missions pursuing SMD PSD's scientific goals.

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Three sets of sufficient conditions for a weak minimum are derived for a nonsingular form of the Bolza problem of variational calculus, expressed in control notation. Both endpoints are assumed to be variable, subject to a set of equality constraints. The end conditions are assumed to be separated. Both specified and unspecified end times are considered. Conditions involving the backward and/or forward integration of a matrix Riccati equation are obtained. Several of these conditions are related to classical conditions involving conjugate points or focal points, but are more easily implemented in realistic optimization problems. Other conditions appear to have no direct classical counterparts. Necessary conditions for a weak minimum differ only slightly from the sufficient conditions. These conditions are easier to apply than another recently proposed set of second-order necessary conditions. The results of this paper are applied to two simple problems.Introduction THE form of the Bolza problem considered in this paper may be expressed in modern control notation in the following manner.Among the set of all continuous unbounded n-dimensional state variable functions x(t) and m-dimensional control variable functions u(t) satisfying nonlinear differential equations of the formas well as r 0 < n initial constraints of the formand rf < n terminal constraints of the formfind the set that will minimize the scalar performance indexProblems of this form are said to have separated end conditions. 1 The initial and terminal times are assumed to be unspecified. y 0 and yf are vectors of constants. The functions /, L, Y, and g are assumed to be twice continuously differentiable with respect to their arguments. A dot above a variable denotes differentiation with respect to time. The subscripts o and / denote evaluation of a quantity at t 0 and fy, respectively. Many optimization probDownloaded by UNIVERSITY OF MICHIGAN on February 20, 2015 | http://arc.aiaa.org | SECOND-ORDER OPTIMALITY CONDITIONS FOR THE BOLZA PROBLEM 213lems for deterministic, nonlinear, time-varying systems are expressible in this manner. Problems of this sort are considered in the classical literature on variational calculus. In Ref. 1 Bliss demonstrates, using the classical notation, that a normal arc without corners which satisfies the system equations (1) and endpoint constraints (2, 3) is locally minimizing only if the multiplier rule, the Weierstrass condition, the Clebsch condition, and a fourth necessary condition are satisfied. This set of necessary conditions is then shown to be sufficient for a local minimum if the last three conditions are suitably strengthened. The fourth necessary (sufficient) condition is that the second variation of J be nonnegative (positive definite) along a minimizing arc. Variants of this fourth condition which are discussed by Bliss include conditions involving conjugate points and characteristic numbers of a boundary -value problem associated with the second variation. None of these conditions is easily implemented, h...