-null controllability for the heat equation and its consequences for the time optimal control problem. (English) Zbl 1165.93016
Summary: We establish a certain -null controllability for the internally controlled heat equation in , with the control restricted to a product set of an open nonempty subset in and a subset of positive measure in the interval . Based on this, we obtain a bang-bang principle for the time optimal control of the heat equation with controls taken from the set measurable; a.e. in , where is a closed and bounded subset of . Namely, each optimal control of the problem satisfies necessarily the bang-bang property: for almost all , where denotes the boundary of the set and is the optimal time. We also get the uniqueness of the optimal control when the target set is convex and the control set is a closed ball.
|93C35||Multivariable systems, multidimensional control systems|
|93C05||Linear control systems|
|49J30||Optimal solutions belonging to restricted classes (existence)|