Optimal control problems with algebraic constraint

Based on a multiple shooting method for parameter identification in differential-algebraic equations due to Heim [4], a new implementation of a direct multiple shooting method for optimal control problems has been developed, which enables the solution of problems that can be separated into different phases. In each of these phases, which might be of unknown length, the control behavior due to inequality constraints, the differential equations, even the dimensions of the state and/or the control space can differ. For the optimal control problems under investigation, the different phases are concerned with the different steps of the recipes. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

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