Time optimal control problems for an internally controlled heat equation with pointwisecontrol constraints are studied. By Pontryagin’s maximum principle and properties ofnontrivial solutions of the heat equation, we derive a bang-bang property for time optimalcontrol. Using the bang-bang property and establishing certain connections between timeand norm optimal control problems for the heat equation, necessary and sufficientconditions for the optimal time and the optimal control are obtained.

An approximation procedure for time optimal control problems for the linear wave equation is analyzed. Its asymptotic behavior is investigated and an optimality system including the maximum principle and the transversality conditions for the regularized and unregularized problems are derived.

We consider a time optimal control problem arisen from the optimalmanagement of a bioreactor devoted to the treatment ofeutrophicated water. We formulate this realistic problem as astate-control constrained time optimal control problem. Afteranalyzing the state system (a complex system of coupled partialdifferential equations with non-smooth coefficients foradvection-diffusion-reaction with Michaelis-Menten kinetics,modelling the eutrophication processes) we demonstrate theexistence of, at least, an optimal solution. Then we present adetailed derivation of a first order optimality condition(involving the corresponding adjoint systems) characterizing theseoptimal solutions. Finally, a numerical example is shown.