Abstract
In this paper, we study two subjects on internally controlled heat equations with time varying potentials: the attainable subspaces and the bang-bang property for some time optimal control problems. We present some equivalent characterizations on the attainable subspaces and provide a sufficient condition to ensure the bang-bang property. Both the above-mentioned characterizations and the sufficient condition are closely related to some function spaces consisting of some solutions to the adjoint equations. It seems for us that the existing ways to derive the bang-bang property for heat equations with time-invariant potentials (see, for instance, [H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, Elsevier, Amsterdam 2005], [F. Gozzi and P. Loreti, SIAM J. Control Optim., 37 (1999), pp. 1195--1221], [V. Mizel and T. Seidman, SIAM J. Control Optim., 35 (1997), pp. 1204--1216], and [G. Wang, SIAM J. Control Optim., 47 (2008), pp. 1701--1720]) do not work for the case where the potentials are time varying. We provide another way to approach it in the current paper.
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