Convergence rates of symplectic Pontryagin approximations in optimal control theory

Sandberg, Mattias; Szepessy, Anders

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 149-173.

Many inverse problems for differential equations can be formulated as optimal control problems. It is well known that inverse problems often need to be regularized to obtain good approximations. This work presents a systematic method to regularize and to establish error estimates for approximations to some control problems in high dimension, based on symplectic approximation of the hamiltonian system for the control problem. In particular the work derives error estimates and constructs regularizations for numerical approximations to optimally controlled ordinary differential equations in ${\Re }^d$, with non smooth control. Though optimal controls in general become non smooth, viscosity solutions to the corresponding Hamilton-Jacobi-Bellman equation provide good theoretical foundation, but poor computational efficiency in high dimensions. The computational method here uses the adjoint variable and works efficiently also for high dimensional problems with $d\gg 1$. Controls can be discontinuous due to a lack of regularity in the hamiltonian or due to colliding backward paths, i.e. shocks. The error analysis, for both these cases, is based on consistency with the Hamilton-Jacobi-Bellman equation, in the viscosity solution sense, and a discrete Pontryagin principle: the bi-characteristic hamiltonian ODE system is solved with a ${𝒞}^2$ approximate hamiltonian. The error analysis leads to estimates useful also in high dimensions since the bounds depend on the Lipschitz norms of the hamiltonian and the gradient of the value function but not on $d$ explicitly. Applications to inverse implied volatility estimation, in mathematical finance, and to a topology optimization problem are presented. An advantage with the Pontryagin based method is that the Newton method can be applied to efficiently solve the discrete nonlinear hamiltonian system, with a sparse jacobian that can be calculated explicitly.

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