Keywords: active structural control under stochastic uncertainty, optimal regulators, stochastic optimal open-loop feedback control, stochastic Hamiltonian, H-minimum control, two-point boundary value problems.

Active regulator strategies are considered [1] for stabilizing dynamic mechanical structures under stochastic applied loadings. The problem is modeled in the framework of stochastic optimal control for minimizing the expected total costs arising from the displacements of the structure and the regulation costs. Due to the great advantages of open-loop feedback controls, stochastic optimal open-loop feedback controls are constructed by taking into account the random parameter variations in the stochastic structural control problem. For finding first stochastic optimal open-loop controls, on the remaining time intervals t_b<=t<=t_f with t₀<=t_b<=t_f, the stochastic Hamilton function of the control problem is considered. Then, the class of H-minimum controls can be determined by solving a finite-dimensional stochastic optimization problem [2] for minimizing the conditional expectation of the stochastic Hamiltonian subject to the remaining deterministic control constraints at each time point t. Having a H-minimum control, the related two-point boundary value problem with random parameters is formulated for the computation of the stochastic optimal state and adjoint state trajectory. As a result of the linear-quadratic structure of the underlying control problem, the state and adjoint state trajectory can be determined analytically to a large extent. Inserting then these trajectories into the H-minimum control, stochastic optimal open-loop controls are found on an arbitrary remaining time interval. These controls then immediately yield a stochastic optimal open-loop feedback control law.

1

C. Block, "Aktive Minderung personeninduzierter Schwingungen an weit gespannten Strukturen im Bauwesen", Fortschrittberichte VDI, Reihe 11, Schwingungstechnik, Nr. 336, VDI-Verlag GmbH, Düsseldorf, 2008.

2

K. Marti, "Stochastic Optimization Problems", 2nd edition, Springer-Verlag, Berlin-Heidelberg, 2008.

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