On time-minimal distributed control of vibrating systems governed by an abstract wave equation
The author considers the time-minimal (or time-optimal) problem for a system described by the abstract wave equation \(y_{tt}+Ay(t)=f(t)\), where the trajectory y(t) and the control f(t) take values in a Hilbert space H and A is a positive definite self-adjoint operator in H. The control is assumed to satisfy (1) \(\int^{T}_{0}| f(t)|^ 2dt\leq C^ 2\), where C is given in advance (the trajectory initiates at time 0 and T is the arrival time to the target). The author shows that the system can be driven to equilibrium in finite time; as a consequence, a unique optimal control exists. The main result is that the time-optimal control has least \(L^ 2\) norm in the interval (0,T) (T the minimum time), the minimum norm being C, the constant in (1). This result is computationally important, since (via an initial estimation of the optimal time) it reduces a ''hard constraint'' problem such as the time- optimal problem to a ''soft constraint'' problem, much easier to deal with in actual computation.
scientific article; zbMATH DE number 3909191