Generalized Euler-Lagrange Equation: A Challenge to Schwartz’s Distribution Theory

An interpretation is given for the Euler-Lagrange equation of optimal control, when the latter contains singularities. This is the case when in an optimal control problem the dynamics and/or the performance index has discontinuities with respect to the state variables. While in each domain of continuity, the necessary conditions for optimality are easily established, their interpretation at the boundaries between domains was not well-understood. In particular, these singularities cannot be dealt with using Schwartz’s distribution theory. In order to make sense of the Euler-Lagrange equation at a domain boundary, one needs to transcend the classical theory. Suitable extensions must allow for the questionable behavior of impulses multiplied by discontinuities, and the notion of partial derivatives at a discontinuity. Using notions from Nonstandard Analysis (NSA), some simple examples illustrate the feasibility and utility of this development.