Keywords: stochastic optimal control, robust regulators, stochastic Hamiltonian, H-minimum control, two-point boundary value problems.

In order to stabilize mechanical structures under dynamic applied loads, active control strategies are taken into account. The structures usually are stationary, safe and stable without external dynamic disturbance. Thus, in case of dynamic disturbance, additional control elements can be installed enabling active control actions.

Active control strategies for mechanical structures are applied in order to counteract heavy applied dynamic loads, such as earthquakes, wind turbulence, water waves, etc., which would lead to large vibrations causing possible damage to the structure. Modeling the structural dynamics by means of a system of first order random differential equations for the state vector (displacement vector q and time derivative of q), robust optimal controls are determined in order to cope with the stochastic uncertainty involved in the dynamic parameters, the initial values and the applied loadings.

Basically, active control depends on the supply of external energy to counteract the dynamic response of a structure. An active control system consists therefore of i) sensors installed at suitable locations of the structure to measure either external excitations or structural response quantities, ii) devices to process the measured information and to compute necessary control forces based on a given control algorithm, and iii) actuators to produce the required control forces. Possible technical devices, concepts, actuators to realize active structural controls are for example electro-hydraulic servomechanisms, passive/active/hybrid/semi-active damping strategies, viscoelastic dampers, tuned mass dampers, aerodynamic appendages, gas pulse generators, gyroscopes, active structural members and joints.

Active structural enhancement consists of the use of active control to modify structural behavior. This enhancement can be used to actively stiffen, or strengthen (against Euler buckling) a given structure. Therefore, actively controlled structures can adaptively modify their stiffness properties to be either stiff or flexible as demanded. For example, optimal control strategies maximize the critical buckling load using sensors and actuators. The aim is then to actively stabilize the structure to prevent it from collapsing.

Large space structure face difficult problems of vibration control. Because they require low weight, such structures will lack the stiffness and damping necessary for the passive control of vibration. Hence current research is directed towards the design of active vibration control to reduce the mean square response, i.e. the displacements, of the system to a desired level within a reasonable span of time.

While the actual time path of the random external load is not known at the planning stage, we may assume that the probability distribution or at least the occurring moments of the applied load are known. The performance of the stochastic dynamic system is evaluated by means of a convex, quadratic cost function along the trajectory: Costs for displacements and feedback control. The problem is then to determine an optimal feedback control law minimizing the expected total costs.

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