Optimization of hybrid dynamic systems typically involves characterizing switching times and mode sequences. Operating autonomous hybrid systems with varying event timings presents multiple challenges. Complexity often arises from optimal mode sequence determination, making optimization of the corresponding hybrid automaton more difficult. An experimental autonomous hybrid dynamic system experiencing switching due to choked flow conditions is presented. Implementing gradient-based optimization algorithms may be difficult due to complex switching patterns of the hybrid dynamic systems. To mitigate this, an approximation of the Heaviside step function is applied to transform the hybrid switching functions to a continuous and smooth forms. Using the Control Vector Parameterization (CVP) approach and evaluating gradients using the variational method, an open-loop time-optimal control problem is presented. Modeling and control methods are demonstrated using an experimental two-tank air system. The optimal control formulation can include constraints to avoid specific undesirable modes of operation.