This paper presents an algorithm to solve nonconvex optimal control problems, where non-convexity can arise from nonlinear dynamics, and non-convex state and control constraints. This paper assumes that the state and control constraints are already convex or convexified, the proposed algorithm convexifies the nonlinear dynamics, via a linearization, in a successive manner. Thus at each succession, a convex optimal control subproblem is solved. Since the dynamics are linearized and other constraints are convex, after a discretization, the subproblem can be expressed as a finite dimensional convex programming subproblem. Since convex optimization problems can be solved very efficiently, especially with custom solvers, this subproblem can be solved in time-critical applications, such as real-time path planning for autonomous vehicles. Several safe-guarding techniques are incorporated into the algorithm, namely virtual control and trust regions, which add another layer of algorithmic robustness. A convergence analysis is presented in continuoustime setting. By doing so, our convergence results will be independent from any numerical schemes used for discretization. Numerical simulations are performed for an illustrative trajectory optimization example.

This paper presents a Successive Convexification (SCvx) algorithm to solve a class of non-convex optimal control problems with certain types of state constraints. Sources of nonconvexity may include nonlinear dynamics and non-convex state/control constraints. To tackle the challenge posed by non-convexity, first we utilize exact penalty function to handle the nonlinear dynamics. Then the proposed algorithm successively convexifies the problem via a project-and-linearize procedure. Thus a finite dimensional convex programming subproblem is solved at each succession, which can be done efficiently with fast Interior Point Method (IPM) solvers. Global convergence to a local optimum is demonstrated with certain convexity assumptions, which are satisfied in a broad range of optimal control problems. The proposed algorithm is particularly suitable for solving trajectory planning problems with collision avoidance constraints. Through numerical simulations, we demonstrate that the algorithm converges reliably after only a few successions. Thus with powerful IPM based custom solvers, the algorithm can be implemented onboard for real-time autonomous control applications.

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