Optimal Control of Structural Dynamic Systems in One Space Dimension Using a Maximum Principle

A maximum principle is developed for a class of problems involving the optimal control of a system of linear hyperbolic equations in one space dimension that are not necessarily separable. An index of performance is formulated, which consists of vector functions of the state variable, their first- and second-order space derivatives and first-order time derivative, and a penalty function involving the open-loop control force vector. The solution of the optimal control problem can easily be shown to be unique using convexity arguments. The given maximum principle involves a Hamiltonian, which contains an adjoint vector function as well as an admissible control vector function. The maximum principle can be used to compute the optimal control vector function and is particularly suitable for problems involving the active control of structural elements for vibration suppression. A numerical example is given which studies the active control of a beam undergoing flexural and torsional vibrations. A comparison of the energies of controlled and uncontrolled beams indicates that the proposed control method is quite effective in damping out the vibrations of structural systems.