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Approximation of Optimally Controlled Ordinary and Partial Differential Equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0003-2669-359X
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.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 06:03
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4066ISBN: 91-7178-412-8 (print)OAI: oai:DiVA.org:kth-4066DiVA: diva2:10612
Public defence
2006-08-30, Sal F3, Lindstedtsvägen 12, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20100917Available from: 2006-07-21 Created: 2006-07-21 Last updated: 2011-12-14Bibliographically approved
List of papers
1. Convergence rates of symplectic pontryagin approximations in optimal control theory
Open this publication in new window or tab >>Convergence rates of symplectic pontryagin approximations in optimal control theory
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
Keyword
optimal control, Hamilton-Jacobi, Hamiltonian system, Pontryagin principle
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-6029 (URN)10.1051/m2an:2006002 (DOI)000235837500007 ()
Note
QC 20100906Available from: 2006-07-21 Created: 2006-07-21 Last updated: 2011-12-20Bibliographically approved
2. Convergence rates for an optimally controlled ginzburg-landau equation
Open this publication in new window or tab >>Convergence rates for an optimally controlled ginzburg-landau equation
2008 (English)Article in journal (Other academic) Submitted
Abstract [en]

An optimal control problem related to the probability oftransition between stable states for a thermally driven Ginzburg-Landauequation is considered. The value function for the optimal control problemwith a spatial discretization is shown to converge quadratically tothe value function for the original problem. This is done by using thatthe value functions solve similar Hamilton-Jacobi equations, the equationfor the original problem being defined on an infinite dimensionalHilbert space. Time discretization is performed using the SymplecticEuler method. Imposing a reasonable condition this method is shownto be convergent of order one in time, with a constant independent ofthe spatial discretization.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-6030 (URN)
Note
QS 20120315Available from: 2006-07-21 Created: 2006-07-21 Last updated: 2012-03-15Bibliographically approved
3. Symplectic Pontryagin Approximations for Optimal Design
Open this publication in new window or tab >>Symplectic Pontryagin Approximations for Optimal Design
2009 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, Vol. 43, no 1, 3-32 p.Article in journal (Refereed) Published
Abstract [en]

The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the Hamiltonian; next the solution to its stationary Hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the Hamiltonian function can be explicitly formulated and when the Jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L 2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems.

Place, publisher, year, edition, pages
EDP Sciences, 2009
Keyword
Topology optimization; inverse problems; Hamilton-Jacobi; regularization; error estimates; impedance tomography; convexification; homogenization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-6071 (URN)10.1051/m2an/2008038 (DOI)000262735400002 ()2-s2.0-59049090036 (Scopus ID)
Note
QC 20100712. Uppdaterad från accepted till published (20100712).Available from: 2008-10-21 Created: 2008-10-21 Last updated: 2011-12-20Bibliographically approved
4. Convergence of the forward euler method for nonconvex differential inclusions
Open this publication in new window or tab >>Convergence of the forward euler method for nonconvex differential inclusions
2008 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 47, no 1, 308-320 p.Article in journal (Refereed) Published
Abstract [en]

The convergence of reachable sets for nonconvex differential inclusions is considered. When the right-hand side in the differential inclusion is a compact-valued, Lipschitz continuous set-valued function it is shown that the convergence in Hausdorff distance of reachable sets for a forward Euler discretization is linear in the time step.

Keyword
differential inclusion, forward Euler, convergence order, convexification
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-6032 (URN)10.1137/070686093 (DOI)000263103800015 ()
Note
QC 20100917Available from: 2006-07-21 Created: 2006-07-21 Last updated: 2010-09-17Bibliographically approved

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