Optimal Control of Partial Differential-Algebraic Equations

Hannes Meinlschmidt* and Stefan Ulbrich

We consider optimal control problems constrained by Partial Differential-Algebraic Equations (PDAEs). There exist various (valid) opinions on what a PDAE is supposed to be. We follow the classical point of view from analysis: in the same way as one may consider a PDE as an abstract ODE on a suitable Banach space, we consider PDAEs as abstract DAEs on suitable Banach spaces. In this talk, we give a short survey about the idea of an DAE and its index and define what we call a PDAE and a PDAE-constrained optimal control problem. Since arbitrary nonlinear PDAEs are, of course, too general to establish a satisfying theory, we concentrate on the so-called semi-explicit PDAEs of index one or two. These correspond to (possibly nonlinearly coupled) systems of PDEs of mixed type (e.g. parabolic-elliptic). Here, we consider a general framework for a large class of problems and explore the usual optimality theory. Moreover, we show that there are popular and interesting examples present in the class of semi-explicit PDAEs of index one or two, and discuss the correspondence of the PDAE index and the underlying model dynamics.

Mathematics Subject Classification: 49K20 49K27 35M33 49J50

Keywords: optimal control; partial differential-algebraic equations

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