This investigation presents a methodology that simplifies the design of multiphase trajectories for aerospace vehicles using indirect methods. Such systems can be viewed as autonomously switched hybrid systems. The trajectories for such systems are subject to piecewise continuous dynamic equations. Moreover, each phase can be associated with a unique cost functional. Therefore, the cost functional of the overall trajectory design problem is also piecewise continuous. Consequently, the necessary conditions of optimality result in a multipoint boundary value problem in a system of differential-algebraic equations. These equations can be difficult to solve because the convergence of existing numerical algorithms is contingent upon supplying an initial guess that is close to the solution, which is not straightforward. The proposed method addresses this limitation, in part, by reducing the problem to a two-point boundary value problem by introducing sigmoid functions to obtain a single system of dynamic equations and a cost functional that are continuous and differentiable throughout the multiphase trajectory. The resultant reduced problem is solved by employing a continuation scheme, wherein the solution approach begins with solving a trivial problem, which is then evolved iteratively to the problem of interest. The proposed method is demonstrated using two examples: 1) time-switched Atlas V 411 launch to circular orbit; and 2) state-switched Mars entry, descent, and landing. The resultant trajectories are found to approximate the solutions of the respective original multipoint boundary value problems very well.
 , “Differential Automata and Their Discrete Simulators,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 11, No. 6, 1987, pp. 665–683. doi:https://doi.org/10.1016/0362-546X(87)90034-4
 , “Optimality Zone Algorithms for Hybrid Systems: Efficient Algorithms for Optimal Location and Control Computation,” Hybrid Systems: Computation and Control, Vol. 3927, Lecture Notes in
Computer Science Series, Springer, New York, 2006, pp. 123–137. doi:https://doi.org/10.1007/11730637_12
 , “Optimal Control of Switching Systems,” Automatica, Vol. 41, No. 1, 2005, pp. 11–27. doi:https://doi.org/10.1016/j.automatica.2004.08.003 ATCAA9 0005-1098
 , “About Solving Hybrid Optimal Control Problems,” 17th IMACS World Congress, 2005.
 , “Transition-Time Optimization for Switched-Mode Dynamical Systems,” IEEE Transactions on Automatic Control, Vol. 51, No. 1, 2006, pp. 110–115. doi:https://doi.org/10.1109/TAC.2005.861711 IETAA9 0018-9286
 , “Optimal Start-Up of an Evaporation System Modeled as an Interconnected Hybrid Dynamical System,” M.S. Dissertation, School of Electrical and Computer Engineering, Purdue Univ., West Lafayette, IN, May 2015.
 , “Hybrid Model Predictive Control for Stabilization of Wheeled Mobile Robots Subject to Wheel Slippage,” IEEE International Conference on Robotics and Automation, IEEE Publ., Piscataway, NJ, Dec. 2007, pp. 2373–4378. doi:https://doi.org/10.1109/ROBOT.2007.363674
 , “Automated Design of Multiphase Space Missions Using Hybrid Optimal Control,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 5, 2013, pp. 1410–1424. doi:https://doi.org/10.2514/1.58766 JGCODS 0731-5090
 , “On the Hybrid Optimal Control Problem: Theory and Algorithms,” IEEE Transactions on Automatic Control, Vol. 52, No. 9, 2007, pp. 1587–1603. doi:https://doi.org/10.1109/TAC.2007.904451 IETAA9 0018-9286
 , “Correction to ‘On the Hybrid Optimal Control Problem: Theory and Algorithms’,” IEEE Transactions on Automatic Control, Vol. 54, No. 6, 2009, p. 1440. doi:https://doi.org/10.1109/TAC.2009.2015528 IETAA9 0018-9286
 , “Optimal Control of Switched Systems Based on Parameterization of the Switching Instants,” IEEE Transactions on Automatic Control, Vol. 49, No. 1, 2004, pp. 2–16. doi:https://doi.org/10.1109/TAC.2003.821417 IETAA9 0018-9286
 , “An Algorithm for Discrete State Sequence and Trajectory Optimization for Hybrid Systems with Partitioned State Space,” 49th IEEE Conference on Decision and Control, IEEE Publ., Piscataway, NJ, Dec. 2010, pp. 4223–4229. doi:https://doi.org/10.1109/CDC.2010.5717264
 , “Optimal Low-Thrust Orbital Maneuvers via Indirect Swarming Method,” Journal of Optimization Theory and Applications, Vol. 162, No. 1, 2014, pp. 272–292. doi:https://doi.org/10.1007/s10957-013-0471-9 JOTABN 0022-3239
 , “Optimization Problems for Dynamic Systems with Path Constraints,” Applied Optimal Control: Optimization, Estimation and Control, Revised Printing, Hemisphere, Washington, D.C., 1975, pp. 90–127.
 , “The Euler-Lagrange Theorem,” Optimal Control with Aerospace Applications, Springer, El Segundo, CA, 2014, pp. 52–56. doi:https://doi.org/10.1007/978-1-4614-8945-0
 , “Classical Fixed Endpoint Problems,” Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966, pp. 87–92.
 , “Initial-Value Methods (Shooting),” Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell, Waltham, MA, 1968, pp. 39–71.
 , “Finite Difference Methods,” Numerical Solution of Boundary Value Problems of Ordinary Differential Equations, Prentice–Hall, Upper Saddle River, NJ, 1988, pp. 185–274. doi:https://doi.org/10.1137/1.9781611971231
 , “A BVP Solver Based on Residual Control and the Matlab PSE,” ACM Transactions on Mathematical Software, Vol. 27, No. 3, 2001, pp. 299–316. doi:https://doi.org/10.1145/502800.502801 ACMSCU 0098-3500
 , “Optimization Problems for Dynamic Systems,” Applied Optimal Control: Optimization, Estimation and Control, Revised Printing, Hemisphere, Washington, D.C., 1975, pp. 42–89.
 , “Rapid Indirect Trajectory Optimization for Conceptual Design of Hypersonic Missions,” Journal of Spacecraft and Rockets, Vol. 52, No. 1, 2015, pp. 177–182. doi:https://doi.org/10.2514/1.A32949 JSCRAG 0022-4650
 Atlas V Launch Services User’s Guide, United Launch Alliance, Centennial, CO, March 2010, pp. 1–5.
 , “A Family of Embedded Runge-Kutta Formulae,” Journal of Computational and Applied Mathematics, Vol. 6, No. 1, 1980, pp. 19–26. doi:https://doi.org/10.1016/0771-050X(80)90013-3 JCAMDI 0377-0427
 , “Mars Science Laboratory: Entry, Descent, and Landing System Performance,” IEEE Aerospace Conference, IEEE Paper 2007-1467, Piscataway, NJ, March 2007, p. 1467.
 , “Assessment of the Mars Science Laboratory Entry, Descent, and Landing Simulation,” American Astronautical Soc. Paper 13-420, Feb. 2013.
 , “Monopropellant Hydrazine 700 lbf Throttling Terminal Descent Engine for Mars Science Laboratory,” 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, AIAA Paper 2007-5481, July 2007. doi:https://doi.org/10.2514/6.2007-5481
 , “Verification and Validation of the Mars Science Laboratory/Curiosity Rover Entry, Descent, and Landing System,” Journal of Spacecraft and Rockets, Vol. 51, No. 4, 2014, pp. 1251–1269. doi:https://doi.org/10.2514/1.A32680 JSCRAG 0022-4650
 , “Seven Minutes of Terror and ‘Some Science’,” 12th International Planetary Probe Workshop, Cologne, Germany, 2015, https://solarsystem.nasa.gov/docs/4_05_Seven%20Minutes%20of%20Terror%20and%20Some%20Science_S. %20Saikia.pdf [retrieved 12 March 2017].
 , “A General Stagnation Point Convective Heating Equation for Arbitrary Gas Mixtures,” NASA TR-R-376, Nov. 1971.