861 research outputs found
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations
Combining recent moment and sparse semidefinite programming (SDP) relaxation
techniques, we propose an approach to find smooth approximations for solutions
of problems involving nonlinear differential equations. Given a system of
nonlinear differential equations, we apply a technique based on finite
differences and sparse SDP relaxations for polynomial optimization problems
(POP) to obtain a discrete approximation of its solution. In a second step we
apply maximum entropy estimation (using moments of a Borel measure associated
with the discrete solution) to obtain a smooth closed-form approximation. The
approach is illustrated on a variety of linear and nonlinear ordinary
differential equations (ODE), partial differential equations (PDE) and optimal
control problems (OCP), and preliminary numerical results are reported
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
We consider the problem of certifying an inequality of the form ,
, where is a multivariate transcendental function, and
is a compact semialgebraic set. We introduce a certification method, combining
semialgebraic optimization and max-plus approximation. We assume that is
given by a syntaxic tree, the constituents of which involve semialgebraic
operations as well as some transcendental functions like , ,
, etc. We bound some of these constituents by suprema or infima of
quadratic forms (max-plus approximation method, initially introduced in optimal
control), leading to semialgebraic optimization problems which we solve by
semidefinite relaxations. The max-plus approximation is iteratively refined and
combined with branch and bound techniques to reduce the relaxation gap.
Illustrative examples of application of this algorithm are provided, explaining
how we solved tight inequalities issued from the Flyspeck project (one of the
main purposes of which is to certify numerical inequalities used in the proof
of the Kepler conjecture by Thomas Hales).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the
European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250,
copyright EUCA 201
Certification of Bounds of Non-linear Functions: the Templates Method
The aim of this work is to certify lower bounds for real-valued multivariate
functions, defined by semialgebraic or transcendental expressions. The
certificate must be, eventually, formally provable in a proof system such as
Coq. The application range for such a tool is widespread; for instance Hales'
proof of Kepler's conjecture yields thousands of inequalities. We introduce an
approximation algorithm, which combines ideas of the max-plus basis method (in
optimal control) and of the linear templates method developed by Manna et al.
(in static analysis). This algorithm consists in bounding some of the
constituents of the function by suprema of quadratic forms with a well chosen
curvature. This leads to semialgebraic optimization problems, solved by
sum-of-squares relaxations. Templates limit the blow up of these relaxations at
the price of coarsening the approximation. We illustrate the efficiency of our
framework with various examples from the literature and discuss the interfacing
with Coq.Comment: 16 pages, 3 figures, 2 table
Semidefinite Relaxations for Stochastic Optimal Control Policies
Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation
have led to the discovery of a formulation of the value function as a linear
Partial Differential Equation (PDE) for stochastic nonlinear systems with a
mild constraint on their disturbances. This has yielded promising directions
for research in the planning and control of nonlinear systems. This work
proposes a new method obtaining approximate solutions to these linear
stochastic optimal control (SOC) problems. A candidate polynomial with variable
coefficients is proposed as the solution to the SOC problem. A Sum of Squares
(SOS) relaxation is then taken to the partial differential constraints, leading
to a hierarchy of semidefinite relaxations with improving sub-optimality gap.
The resulting approximate solutions are shown to be guaranteed over- and
under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in
Portland, Oregon. 7 pages, colo
Tropical Kraus maps for optimal control of switched systems
Kraus maps (completely positive trace preserving maps) arise classically in
quantum information, as they describe the evolution of noncommutative
probability measures. We introduce tropical analogues of Kraus maps, obtained
by replacing the addition of positive semidefinite matrices by a multivalued
supremum with respect to the L\"owner order. We show that non-linear
eigenvectors of tropical Kraus maps determine piecewise quadratic
approximations of the value functions of switched optimal control problems.
This leads to a new approximation method, which we illustrate by two
applications: 1) approximating the joint spectral radius, 2) computing
approximate solutions of Hamilton-Jacobi PDE arising from a class of switched
linear quadratic problems studied previously by McEneaney. We report numerical
experiments, indicating a major improvement in terms of scalability by
comparison with earlier numerical schemes, owing to the "LMI-free" nature of
our method.Comment: 15 page
Natural preconditioners for saddle point systems
The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or discrete setting, so saddle point systems arising from discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as does, for example, the widely used sequential quadratic programming approach to nonlinear optimization.\ud
This article concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee rapid convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness -- in terms of rapidity of convergence -- is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends
- …