Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems
We consider a quadratic optimal control problem governed by a nonautonomous affine ordinary differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we provide a first order expansion for the penalized states and adjoint states around the state and adjoint state of the original problem. Our main argument relies on the following fact: if the optimal control satisfies strict complementarity conditions for its Hamiltonian except for a set of times with null Lebesgue measure, the functional estimates for the penalized optimal control problem can be derived from the estimates of a related finite dimensional problem. Our results provide several types of efficiency measures of the penalization technique: error estimates of the control for L (s) norms (s in [1, +a]), error estimates of the state and the adjoint state in Sobolev spaces W (1,s) (s in [1, +a)) and also error estimates for the value function. For the L (1) norm and the logarithmic penalty, the sharpest results are given, by establishing an error estimate for the penalized control of order where is the (small) penalty parameter.
optimal control interior-point algorithms sensitivity expansion of value function and solutions