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The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Started from the first-order necessary conditions for optimality, a Hamiltonian system is derived for the state and adjoints with split boundary conditions. Then, with recognition of the two-point boundary-value problem as a canonical transformation, generating functions are employed to find the optimal feedback control, as well as the optimal trajectory. Although the optimal control problem is formulated in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a nonlinear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two-point boundary-value problem using a forward-shooting method. Nomenclature A= linearized system dynamics about the circular reference trajectory arg min (·) = argument minimum with respect to the variable (·) B = linearized system dynamics about the circular reference trajectory F = nongravitational control force vector F 1 , F 2 , F 3 , F 4 = principal kinds of generating functions f (x, u, t) = system dynamics H (x, p, u, t) = Hamiltonian of the system I = identity matrix i = radial unit vector in the rotating coordinate frame J = performance index to be minimized j = tangential unit vector in the rotating coordinate frame k = normal unit vector in the rotating coordinate frame L(x, u, t) = full time performance index or Lagrangian m = mass of the spacecraft, assumed constant in the current application p = adjoint vector R = position vector of the origin of the rotating frame from the central body, Ri r = position vector of the spacecraft from the center of gravity r = |r| t = time u = control vector IntroductionT HIS paper presents a novel approach to evaluating optimal continuous thrust trajectories and feedback control laws for a spacecraft subject to a general gravity field. This approach is derived by relying on the Hamiltonian nature of the necessary conditions associated with optimal control and by using certain properties of generating functions and canonical transformations. In particular, we show that certain solutions to the Hamilton-Jacobi (HJ) equation, associated with canonical transformations of Hamiltonian systems, can directly yield optimal control laws for a general system. Typically, application of Pontryagin's principle changes the nonlinear optimal rendezvous problem to a two-point boundary value problem (TPBVP), for which one generally requires an initial estimate for the initial (or final) adjoint variables fol...

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Started from the first-order necessary conditions for optimality, a Hamiltonian system is derived for the state and adjoints with split boundary conditions. Then, with recognition of the two-point boundary-value problem as a canonical transformation, generating functions are employed to find the optimal feedback control, as well as the optimal trajectory. Although the optimal control problem is formulated in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a nonlinear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two-point boundary-value problem using a forward-shooting method. Nomenclature A= linearized system dynamics about the circular reference trajectory arg min (·) = argument minimum with respect to the variable (·) B = linearized system dynamics about the circular reference trajectory F = nongravitational control force vector F 1 , F 2 , F 3 , F 4 = principal kinds of generating functions f (x, u, t) = system dynamics H (x, p, u, t) = Hamiltonian of the system I = identity matrix i = radial unit vector in the rotating coordinate frame J = performance index to be minimized j = tangential unit vector in the rotating coordinate frame k = normal unit vector in the rotating coordinate frame L(x, u, t) = full time performance index or Lagrangian m = mass of the spacecraft, assumed constant in the current application p = adjoint vector R = position vector of the origin of the rotating frame from the central body, Ri r = position vector of the spacecraft from the center of gravity r = |r| t = time u = control vector IntroductionT HIS paper presents a novel approach to evaluating optimal continuous thrust trajectories and feedback control laws for a spacecraft subject to a general gravity field. This approach is derived by relying on the Hamiltonian nature of the necessary conditions associated with optimal control and by using certain properties of generating functions and canonical transformations. In particular, we show that certain solutions to the Hamilton-Jacobi (HJ) equation, associated with canonical transformations of Hamiltonian systems, can directly yield optimal control laws for a general system. Typically, application of Pontryagin's principle changes the nonlinear optimal rendezvous problem to a two-point boundary value problem (TPBVP), for which one generally requires an initial estimate for the initial (or final) adjoint variables fol...

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