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Convergence rates of symplectic pontryagin approximations in optimal control theory
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0003-2669-359X
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2006 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, Vol. 40, no 1, 149-173 p.Article in journal (Refereed) Published
Abstract [en]

 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 R-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 >> 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.

Place, publisher, year, edition, pages
EDP Sciences, 2006. Vol. 40, no 1, 149-173 p.
Keyword [en]
optimal control, Hamilton-Jacobi, Hamiltonian system, Pontryagin principle
National Category
URN: urn:nbn:se:kth:diva-6029DOI: 10.1051/m2an:2006002ISI: 000235837500007OAI: diva2:10608
QC 20100906Available from: 2006-07-21 Created: 2006-07-21 Last updated: 2011-12-20Bibliographically approved
In thesis
1. Approximation of Optimally Controlled Ordinary and Partial Differential Equations
Open this publication in new window or tab >>Approximation of Optimally Controlled Ordinary and Partial Differential Equations
2006 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

In this thesis, which consists of four papers, approximation of optimal control problems is studied. In Paper I the Symplectic Pontryagin method for approximation of optimally controlled ordinary differential equations is presented. The method consists of a Symplectic Euler time stepping scheme for a Hamiltonian system with a regularized Hamiltonian. Under some assumptions it is shown that the approximate value function associated with this scheme converges to the original value function with a linear rate.

In Paper II the ideas from Paper I are extended to approximation of an optimally controlled partial differential equation, a one-dimensional Ginzburg-Landau equation. The approximation is performed in two steps. In the first step a value function associated with a finite element spatial discretization is shown to converge quadratically in the mesh size to the original value function. In the second step a Symplectic Euler discretization in time is shown to converge with a linear rate. The behavior of optimal solutions is shown by numerical examples.

In Paper III the same approximation method as in Paper II is applied to three other problems; the optimal design of an electric conductor, the design of an elastic domain, and the problem of reconstructing the interior of an object from measured electrical surface currents. Since these problems are time-independent the Hamilton-Jacobi theory can not be used. In order to be able to obtain error bounds the problems are therefore transferred to a setting where time plays a role. Computer experiments with the Symplectic Pontryagin method is performed for all three problems.

Common to the three first papers is that the convergence proofs use that the approximate value functions solve Hamilton-Jacobi equations consistent with the original Hamilton-Jacobi equations.

Paper IV concerns convergence of attainable sets for non-convex differential inclusions. When the right hand side in the differential inclusion is a bounded, Lipschitz set-valued function it is shown that the convergence in Hausdorff-distance of attainable sets for a Forward Euler discretization is linear in the time step. This implies that dynamic programming using Forward Euler discretizations of optimal control problems converge with a linear rate when all the functions involved are bounded and Lipschitz continuous.

Place, publisher, year, edition, pages
Stockholm: KTH, 2006. v, 17 p.
Trita-MAT. MA, ISSN 1401-2278 ; 06:03
National Category
urn:nbn:se:kth:diva-4066 (URN)91-7178-412-8 (ISBN)
Public defence
2006-08-30, Sal F3, Lindstedtsvägen 12, Stockholm, 10:00
QC 20100917Available from: 2006-07-21 Created: 2006-07-21 Last updated: 2011-12-14Bibliographically approved

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