Numerical Optimal Control Via Smooth Penalty Functions
The use of first order necessary conditions to solve optimal control problems has two principal drawbacks: a sufficiently close initial approximation is required to ensure local convergence and this initial approximation must be cHosen so that the convergence is to a local optimum. We present a class of algorithms which resolves both of these difficulties and which is ultimately based on the solution of first order necessary conditions. The key ingredients are three smooth penalty functions (the quadratic penalty for equality constraints and the log barrier or quadratic loss for inequality constraints), a parameterized system of nonlinear equations, and efficient predictor-corrector continuation techniques to follow the penalty path to optimality. This parameterized system of equations is essentially a homotopy which is derived from these penalty functions, contains the penalty path as a solution, and represents a perturbation of the first order necessary conditions. However, it differs significantly from Homotopies for nonlinear equations in that an unconstrained optimization technique is required to obtain an initial point.
KeywordsOptimal Control Problem Penalty Function Sequential Quadratic Programming Continuation Method Pontryagin Maximum Principle
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