About optimal control problem under action duration constraint and infimum-gap
Résumé
We consider optimal control problems with scalar control in [0,1] under the constraint that the length of time during which the control is non-null is bounded by a prescribed value. We show that such problems present an infimum-gap when the solution of the relaxed problem is not bang-bang. Then, we show that the gap can be closed if one considers the infimum over all survival probability functions dominated by the survival probability function of the uniform law as a constraint on the control function.