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Many engineering tasks can be formulated as dynamic optimization or open-loop optimal control problems, where we search a priori for the input profiles to a dynamic system that optimize a given performance measure over a certain time period. Further, many systems of interest in the chemical processing industries experience significant discontinuities during transients of interest in process design and operation. This paper discusses three classes of dynamic optimization problems with discontinuities:  path-constrained problems, hybrid discrete/continuous problems, and mixed-integer dynamic optimization problems. In particular, progress toward a general numerical technology for the solution of large-scale discontinuous dynamic optimization problems is discussed.

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A new method for solving equality path-constrained dynamic optimization problems is proposed that directly solves the associated high-index differential and algebraic equation (DAE) using the dummy derivative method. This method eliminates efficiency and error control problems associated with other numerical methods for solving these problems. It is proved that equality path constraints can cause the dynamic optimization problem to contain high-index DAEs. Also presented are a set of theorems that use controllability arguments to state when the equality path-constrained dynamic optimization problem is feasible. Results from several numerical examples are discussed, including a large-scale chemical engineering example.