This research presents a new method for the numerical solutions to hypersonic trajectory optimization problems. The main contribution is the formulation of highly nonlinear optimal control problem associated with hypersonic trajectory optimization as a sequence of finite-dimensional convex optimization problems, particularly as a sequence of semidefinite programming problems. By introducing extra variables, the original constrained optimal control problem is transformed into a polynomial optimal control problem, which is further converted to a general quadratically constrained quadratic programming problem by introducing more variables and quadratic constraints. A semidefinite relaxation technique is used to relax this nonconvex quadratic programming problem to a linear matrix programming problem with rank-one matrix constraint, which is then solved by a sequential semidefinite programming approach. The convergence of this successive method is proved, which will benefit onboard applications. In each iteration, a sub-problem can be solved by convex algorithms with deterministic convergence properties and prescribed level of accuracy. The maximum impact velocity and maximum downrange trajectory optimization problems for hypersonic flight are solved to verify the effectiveness of this approach by comparing to a sequential quadratic programming algorithm.
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