This work studies the optimal control problem for a class of hybrid systems. First, a model has been chosen to describe the dynamic behavior of the systems considered. In particular, this model focuses on the difference between controlled switchings and autonomous ones. Assumptions are given to ensure the existence and uniqueness of a solution to this model. Then, an optimal control problem is proposed. Again, some sufficient conditions are given on the elements of the cost function to prove the existence of a solution.
The second part of the methodological contribution lies in the numerical computation of a solution to the optimal control problem. Continuous and discrete inputs are parametrized so that the theoritical optimal solution can be approached as close as desired. The continuity and derivability of the cost function versus the decision variables resulting from the parametrization stage are studied. Finally, it is proven that, as in the non hybrid case, sensitivity equations and adjoints can be used to compute efficiently the gradient of the cost function versus the decision variables.
To assess the potential benefit of the proposed method, it has been applied to a solar domestic hot water system. Solar domestic hot water systems are systems which provide hot water owing to solar energy. The on-line solution procedure of the optimal control problem also integrates weather forecasts provided on-line by the Swiss Meteorological Institute (SMI).