Optimal control of higher index DAEs with nondifferentiable cost function

Abstract

The aim of the paper is twofold. First we introduce a class of optimal control problems described by higher index DAEs. We show that these problems can be treated in s similar way to problems described by ordinary ODEs. We formulate two kinds of optimality conditions for the considered class of problems. Each set of necessary optimality conditions could be used to construct algorithms for optimal control problems with higher index DAEs. Secondly, we discuss optimal control problems with nondifferentiable objective functionals. The special case of the considered control problems is a minimum fuel control problems with higher index DAEs. The objective function of this particular problem is represented by the functional: F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> (x,u)=int <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> fSigma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (t,x(t))|u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (t)-u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> (t)|dt+phi(x(1)). In the paper we do not present algorithms for the considered optimal control problems however they would be easily constructed under the assumption that we have at our disposal procedures for integrating higher index DAEs and their corresponding adjoint equations (which are introduced in the paper).