Stochastic Stabilization of Slender Beams in Space: Modeling and Boundary Control K. D. Do and A. D. Lucey Department of Mechanical Engineering, Curtin University, Kent Street, Bentley, WA 6102, Australia Abstract This paper considers the problem of modeling and boundary feedback stabilization of extensible and shearable slender beams with large deformations and large rotations in space under both deterministic and stochastic loads induced by flows. Fully nonlinear equations of motion of the beams are first derived. Boundary feedback controllers are then designed for global practical exponential p-stabilization of the beams based on the Lyapunov direct method. A new Lyapunov-type theorem is developed to study well-posedness and stability of stochastic evolution systems (SESs) in Hilbert space. Key words: Slender beams; Large motions; Boundary control; Stochastic evolution system; Well-posedness; p-stability. 1 Introduction This paper focuses on relatively slender beams, for which the shear magnitude is smaller than that of the spatial gradient of the transverse displacements. Due to their large length-to-diameter ratio, extensibility and shear- ability, the relatively slender beams exhibit both large and small motions (both deflection and rotation) under external (both deterministic and stochastic) loads. Al- though large motions can cause a serious failure (loop formation or hockling) in beams, most of existing bound- ary control works (e.g., [5,6,12,13,17,20,22–25,25,26,28, 38–41, 44, 48, 50] based on the Lyapunov direct and flat- ness methods, and [4, 32, 33] based on the backstepping method on single beams, and [15, 27, 29, 35] on multiple beams) on boundary control have considered only small deflection (vibration). None of the equations of motion in the above works can describe loop formation due to the fact that they are obtained by linearizing the axial stretch and rotational motions, and neglecting the shear strains, see [16, 37], and therefore exclude large motions. Boundary control of slender beams with large motions has received less attention. In [8, 14] (see also [2, 31] for models of slender beams, where only large deflec- tion is considered), boundary control of unshearable ris- ers/beams with large deflections was considered. Bound- ary control of extensible and shearable slender beams has been considered in [9] in three-dimensional space (3D). In these works, the external loads are assumed to be deterministic except for the work in [11], where Email address: duc@curtin.edu.au, T.Lucey@curtin.edu.au (K. D. Do and A. D. Lucey). stochastic external loads are initially considered for slen- der beams in two-dimensional space (2D). Control de- sign and stability analysis for stochastic beams is much harder than for deterministic beams. For example, the stochastic component of flows, which enters to the beam system via the hydrodynamic/aerodynamic Centripetal matrix, potentially destabilizes the beam system under deterministic control designs. The main contributions of this paper consist of three parts. First, fully nonlinear equations of motion of the beams and their properties are derived in an appropri- ate form for boundary control design by using deforma- tion theory and sea loads on offshore structures. The unit quaternion is used for attitude representation of the beams to resolve singularities caused by Euler angles. Second, boundary feedback controllers are designed for global practical exponential stabilization of the beams based on the Lyapunov direct method. In the control design, various Young’s and H¨older’s inequalities and Sobolev embedding, a flexible combination of Earth- fixed and body-fixed coordinates, and cross vector prod- ucts are used. Third, a new Lyapunov-type theorem is developed to study well-posedness and stability analysis of a class of nonlinear SESs in Hilbert space. This theo- rem does not require global monotonic and linear growth conditions as in e.g., [7, 36, 46, 47]. The Lyapunov func- tion uses ∥.∥ V instead of ∥.∥ H as in [12]. This allows to study well-posedness of SESs, for which it is difficult to apply multiple Gelfand triples because V ⊂ H. Notations. The symbols ∧ and ∨ denote the infimum and supremum operators, respectively. The symbols “col”, “×”, “E” denote the column operator, vector cross product operator, and the expected value, respectively. Preprint submitted to Automatica 30 November 2017