Richard Vinter, EEE, Imperial: The Calculus of Variations and Optimization: The Impact of Nonsmooth Analysis
It became apparent, in the late 1950’s, that the classical Calculus of Variations was inadequate to deal with the types of constraints encountered in new areas of potential application, such as engineering design, operations research and mathematical economics. These constraints included dynamic constraints in the form of ‘controlled’ differential equations and path-wise constraints on control and state variables. Optimal Control emerged to fill the gap. In the presence of the new types of constraints, the classical conditions, such as Hamilton’s system of equations and those involving the Hamilton Jacobi equation don’t even make sense, because they involve derivatives of functions that are not differentiable in the conventional sense. A desire to make sense of these earlier conditions and explore their implications, for broader classes of dynamic optimization problems prompted, in turn, the field of Nonsmooth Analysis, which dates from the 1970’s. The idea is to develop useful notions of set-valued derivative, and an accompanying calculus, to give meaning to ‘derivative’ of a general, non-differential function, taking as cue the subddifferential calculus of Convex Analysis. Nonsmooth Analysis has been useful in achieving this goal to the point that it pervades the modern theory of optimal control, with manifestations in viscosity solutions concepts for Hamilton Jacobi equations, and modern versions of the Euler Lagrange condition for dynamic optimizations in which the dynamic constraint takes the form of a differential inclusion. But Nonsmooth Analysis has had an impact further afield. It has extended the scope of Nonlinear Analysis, bringing new proofs and generalizations of mean value theorems, inverse function theorems, etc., and has had ramifications in Optimization and Numerical Analysis.