Differential-Geometric and Nonsmooth Methods in Deterministic Finite-Dimensional Control

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The project deals with research on mathematical problems belonging or related to deterministic, finite-dimensional nonlinear control theory, continuing the principal investigator's previous work in this area on a broad class of theoretical and applied questions, with the objective of solving fundamental open problems in optimal control (in particular on high-order nonsmooth necessary conditions for optimality, and optimal control with state space constraints) that pose significant mathematical challenges, and developing tools with a wide range of potential applications to new problems. The methods used will be those of differential-geometric control theory, especially geometrically intrinsic versions of the Pontryagin Maximum Principle, theories of high-order optimality conditions, and nonsmooth analysis. The basic research strategy will be to use multivalued differentials, flows, and generalized abstract variations, as well as the theory of needle variations for almost lower semicontinuous set-valued maps. Possibly, the theory of real-analytic maps and their associated stratifications will play a role as well.The questions to be studied in this work include large classes of optimization problems that occur in engineering and mechanical applications, such as: (a) control of robotic hands and other robotic systems, which mathematically amounts to path-finding for "non-holonomic systems" (that is, systems where one can directly control only a small number of directions, but one can effectively achieve motion in many other directions as well by combining the basic motions, as in the case of a car, which cannot move sideways but can be parked---as if it could move sideways---by combining forward and backward motions and turns), (b) control of chemical plants, which leads to problems involving so-called "singular controls", (c) control of underwater vehicles (in work now in progress, that has already lead to one publication).