Dynamic programming for free-time problems with endpoint constraints (Q1312291)

The paper considers optimal control problems with free end-time and endpoint constraints aiming at providing necessary and sufficient conditions for local optimality based on a solution to the Hamilton- Jacobi equation under a normality hypothesis. The normality hypothesis requires that first-order necessary conditions of optimality hold in a form where the cost multiplier is nonzero. First, it is shown that if there is a suitable solution to the Hamilton-Jacobi equation (i.e. a verification function) and a nominal state trajectory which satisfies a certain compatibility condition, then that arc must be a minimizer. This is the sufficiency part. Second, it is shown that if \(x(\cdot)\) is a minimizer, then a suitably defined value function is a verification function associated with \(x(\cdot)\). This is the necessity part.