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Abstract—An optimal guidance method is developed that reduces sensitivity to parametric uncertainties in the dynamic model. The method combines a previously developed method for guidance and control using adaptive Legendre-Gauss-Radau (LGR) collocation and a previously developed approach for desensitized optimal control. Guidance updates are performed such that the desensitized optimal control problem is re-solved on the remaining horizon at the start of each guidance cycle. The effectiveness of the method is demonstrated on a simple example using Monte Carlo simulation. The application of the method results in a smaller final state error distribution when compared to desensitized optimal control without guidance as well as a previously developed method for optimal guidance and control. 

A modified form of Legendre-Gauss orthogonal direct collocation is developed for solving optimal control problems whose solutions are nonsmooth due to control discon- tinuities. This new method adds switch time variables, control variables, and collocation conditions at both endpoints of a mesh interval, whereas these new variables and collocation con- ditions are not included in standard Legendre-Gauss orthogonal collocation. The modified Legendre-Gauss collocation method alters the search space of the resulting nonlinear programming problem and optimizes the switch point of the control solution. The transformed adjoint system of the modified Legendre- Gauss collocation method is then derived and shown to satisfy the necessary conditions for optimality. Finally, an example is provided where the optimal control is bang-bang and contains multiple switches. This method is shown to be capable of solving complex optimal control problems with nonsmooth solutions. 

An adaptive mesh refinement method for nu- merically solving optimal control problems is described. The method employs collocation at the Legendre-Gauss-Radau points. Within each mesh interval, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and an adaptive forward- backward explicit integration of the state dynamics. Accuracy in the method is achieved by adjusting the number of mesh intervals and degree of the approximating polynomial in each mesh interval. The method is demonstrated on time-optimal transfers from an L1 halo orbit to an L2 halo orbit in the Earth- Moon system, and performance is compared against previously developed mesh refinement methods. 

A computational framework for the solution of op- timal control problems with time-dependent partial differential equations (PDEs) is presented. The optimal control problem is transformed from a continuous time and space optimal control problem to a sparse nonlinear programming problem through state parameterization with Lagrange polynomials and discrete controls defined at Legendre-Gauss-Radau (LGR) points. The standard LGR collocation method is coupled with a modified Radau method to produce a collocation point on the typically noncollocated boundary. The newly collocated endpoint allows for a representation of the state derivative and control on the originally noncollocated boundary such that Neumann boundary conditions may be satisfied. Finally, the method developed in this paper is demonstrated on a viscous Burgers’ tracking problem and the results are compared to an existing solution. 

A continuation method for solving singular optimal control problems is presented. Assuming that the structure of the optimal solution is known a priori, the time horizon of the optimal control problem is divided into multiple domains and is discretized using a multiple-domain Radau collocation formulation. The resulting nonlinear programming problem is then solved by implementing a continuation method over singular domains. The continuation method is then demonstrated on a minimum-time rigid body reorientation problem. The results obtained demonstrate that a continuation method can be used to obtain an accurate approximation to the optimal control on both a finite-order and an infinite-order singular arc.