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Exact controllability of a string to rest with a moving boundary

Gugat Martin

Control and Cybernetics

2019

Vol. 48, No. 1

69--87

We consider the problem of steering a ﬁnite string to the zero state in ﬁnite time from a given initial state by controlling the state at one boundary point while the other boundary point moves. As a possible application we have in mind the optimal control of a mining elevator, where the length of the string changes during the transportation process. During the transportation process, oscillations of the elevator-cable can occur that can be damped in this way. We present an exact controllability result for Dirichlet boundary control at the ﬁxed end of the string that states that there exist exact controls for which the oscillations vanish after ﬁnite time. For the result we assume that the movements are Lipschitz continuous with a Lipschitz constant, whose absolute value is smaller than the wave speed. In the result, we present the minimal time, for which exact controllability holds, this time depending on the movement of the boundary point. Our results are based upon travelling wave solutions. We present a representation of the set of successful controls that steer the system to rest after ﬁnite time as the solution set of two point-wise equalities. This allows for a transformation of the optimal control problem to a form where no partial diﬀerential equation appears. This representation enables interesting insights into the structure of the successful controls. For example, exact bang-bang controls can only exist if the initial state is a simple function and the initial velocity is zero.