18,809 research outputs found
Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems
We are interested in high-order linear multistep schemes for time
discretization of adjoint equations arising within optimal control problems.
First we consider optimal control problems for ordinary differential equations
and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas
BDF methods preserve high--order accuracy. Subsequently we extend these results
to semi--lagrangian discretizations of hyperbolic relaxation systems.
Computational results illustrate theoretical findings
A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives
We propose a direct numerical method for the solution of an optimal control
problem governed by a two-side space-fractional diffusion equation. The
presented method contains two main steps. In the first step, the space variable
is discretized by using the Jacobi-Gauss pseudospectral discretization and, in
this way, the original problem is transformed into a classical integer-order
optimal control problem. The main challenge, which we faced in this step, is to
derive the left and right fractional differentiation matrices. In this respect,
novel techniques for derivation of these matrices are presented. In the second
step, the Legendre-Gauss-Radau pseudospectral method is employed. With these
two steps, the original problem is converted into a convex quadratic
optimization problem, which can be solved efficiently by available methods. Our
approach can be easily implemented and extended to cover fractional optimal
control problems with state constraints. Five test examples are provided to
demonstrate the efficiency and validity of the presented method. The results
show that our method reaches the solutions with good accuracy and a low CPU
time.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Vibration and Control', available from
[http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised
03-Sept-2018; Accepted 12-Oct-201
Optimal Switching for Hybrid Semilinear Evolutions
We consider the optimization of a dynamical system by switching at discrete
time points between abstract evolution equations composed by nonlinearly
perturbed strongly continuous semigroups, nonlinear state reset maps at mode
transition times and Lagrange-type cost functions including switching costs. In
particular, for a fixed sequence of modes, we derive necessary optimality
conditions using an adjoint equation based representation for the gradient of
the costs with respect to the switching times. For optimization with respect to
the mode sequence, we discuss a mode-insertion gradient. The theory unifies and
generalizes similar approaches for evolutions governed by ordinary and delay
differential equations. More importantly, it also applies to systems governed
by semilinear partial differential equations including switching the principle
part. Examples from each of these system classes are discussed
A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method
In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
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