SOLUTION DIFFERENTIABILITY FOR PARAMETRIC NONLINEAR CONTROL-PROBLEMS WITH CONTROL-STATE CONSTRAINTS

This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameter p of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parametric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corresponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC. Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research.