SUMMARYIt has been common practice to find controls satisfying only necessary conditions for optimality, and then to use these controls assuming that they are (locally) optimal. However, sufficient conditions need to be used to ascertain that the control rule is optimal. Second order sufficient conditions (SSC) which have recently been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii, are a special form of sufficient conditions which are particularly suited for numerical verification. In this paper we present optimization methods and describe a numerical scheme for finding optimal bang-bang controls and verifying SSC. A straightforward transformation of the bang-bang arc durations allows one to use standard optimal control software to find the optimal arc durations as well as to check SSC. The proposed computational verification technique is illustrated on three example applications.

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The paper presents engineering models, optimization algorithms and design results from a Multidisciplinary Design Optimization (MDO) research in the framework of ESA's PRESTIGE PhD program. The application focuses on the conceptual design of classical unmanned Expendable Launch Vehicles, and results are presented from sensitivity studies and validation tests on European launchers (Ariane-5 ECA and VEGA). Relatively simple models and a mixed global/local optimization approach allow obtaining reasonable results with limited computational effort. A critical analysis of the results also leads to the identification of the most critical modeling aspects to be improved to allow for early preliminary design applications. Nomenclature α = engine mixture ratio ε = nozzle expansion ratio θ = pitch angle ψ = yaw angle µ = mean value σ = standard deviation A e = nozzle exhaust area AoA = total angle of attack a = orbit semiaxiscore boosters configuration CpL = cost per launch e = orbit eccentricity GTOW = gross take-off weight i = orbit inclination I sp = specific impulse, nominal conditions (i.e. nozzle optimal expansion) 2 I sp,vac = specific impulse in vacuum I sp,sea = specific impulse at sea level L/D = length over diameter ratio LSP = launch success probability MR = Engine mixture ratio M = Mach number M prop = Propellant mass (usable propellant only) M dry = Dry mass = inert mass + unused propellants mass N s = number of stages N bs = number of booster sets N b,j = number of boosters for j-th boosters set n ax = axial acceleration p cc = chamber pressure PL = payload PLSF = payload scaling factor q dyn = dynamic pressure q heat = heat flux SET = single engine type configuration (i.e. same engine type for all stages) T nom = total thrust, nominal conditions (i.e. nozzle optimal expansion) T ,vac = total thrust in vacuum T sea = total thrust at sea level

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