Algorithms multiple shooting

In the BzzMath library, the multiple shooting algorithms have not been implemented in a dedicated class. 

The free parameters of the model are fitted to experimental data (the 0-I-D-P-fluorescence rise of dark-adapted tobacco leaves) by means of the multiple shooting algorithm PARFIT as developed by Bock [2] for parameter identification in systems of nonlinear differential equations. We use a multiple experiment structure for measurements at different light intensities. The initial trajectory and the results are shown in Figs. 2 and 3. 

Bock, H. G., Plitt, K. J. (1984) A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Proc. of the 9th IFAC Worldcongress, Budapest, Hungary, Vol. IX, Colloquia 14.2, 09.2... 

OC/DO-problems can be solved very efficiently and reliably by combining the above problem discretization (piecewise control parameterization and multiple shooting state discretization) with a specifically tailored sequential quadratic programming (SQP) algorithm for the solution of the large, but structured NLP problem (2.3). Such a strategy has been implemented in the optimal control code MUSCOD [6, 13]. 

In addition, the Jacobian matrix of F2 in (2.3) has a characteristic block-sparse structure due to the special form of the continuity and consistency conditions (2.2). This multiple shooting structure can be exploited by a condensing algorithm to considerably reduce the size of the QP subproblem, which is then solved by a standard QP solver. Alternatively, the original QP subproblem can be directly solved by specialized, large-scale QP solvers, see e.g. [18]. 

H.-G. Bock and K.-J. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. Preprints of the 9th IFAC World Congress, Budapest, International Federation of Automatic Control, 1984. 

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