This paper presents a method to compute optimal open-loop trajectories for systems subject to state and control inequality constraints in which the cost function is quadratic and the state dynamics are linear. For the case in which inequality constraints are decentralized with respect to the controls, optimal Lagrange multipliers enforcing the inequality constraints may be found at any time through Pontryagins minimum principle. In so doing, the set of differential algebraic Euler-Lagrange equations is transformed into a nonlinear two-point boundary-value problem for states and costates whose solution meets the necessary conditions for optimality. The optimal performance of inequality constrained control systems is calculable, allowing for comparison to previous, sub-optimal solutions. The method is applied to the control of damping forces in a vibration isolation system subjected to constraints imposed by the physical implementation of a particular controllable damper. An outcome of this study is the best performance achievable given a particular objective, isolation system, and semi-active damper constraints.
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