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L -null controllability for the heat equation and its consequences for the time optimal control problem. (English) Zbl 1165.93016
Summary: We establish a certain L -null controllability for the internally controlled heat equation in Ω×[0,T], with the control restricted to a product set of an open nonempty subset in Ω and a subset of positive measure in the interval [0,T]. Based on this, we obtain a bang-bang principle for the time optimal control of the heat equation with controls taken from the set 𝒰 ad ={u(·,t):[0,)L 2 (Ω) measurable; u(·,t)U, a.e. in t}, where U is a closed and bounded subset of L 2 (Ω). Namely, each optimal control u * (·,t) of the problem satisfies necessarily the bang-bang property: u * (·,t)U for almost all t[0,T * ], where U denotes the boundary of the set U and T * is the optimal time. We also get the uniqueness of the optimal control when the target set S is convex and the control set U is a closed ball.
MSC:
93B05Controllability
93C35Multivariable systems, multidimensional control systems
93C05Linear control systems
35K05Heat equation
49J30Optimal solutions belonging to restricted classes (existence)