28,134 research outputs found
Control of functional differential equations with function space boundary conditions
Problems involving functional differential equations with terminal conditions in function space are considered. Their application to mechanical and electrical systems is discussed. Investigations of controllability, existence of optimal controls, and necessary and sufficient conditions for optimality are reported
A sufficient optimality condition for delayed state-linear optimal control problems
We give answer to an open question by proving a sufficient optimality
condition for state-linear optimal control problems with time delays in state
and control variables. In the proof of our main result, we transform a delayed
state-linear optimal control problem to an equivalent non-delayed problem. This
allows us to use a well-known theorem that ensures a sufficient optimality
condition for non-delayed state-linear optimal control problems. An example is
given in order to illustrate the obtained result.Comment: This is a preprint of a paper whose final and definite form is with
'Discrete and Continuous Dynamical Systems -- Series B' (DCDS-B), ISSN
1531-3492, eISSN 1553-524X, available at
[http://www.aimsciences.org/journal/1531-3492]. Paper Submitted 31/Dec/2017;
Revised 13/April/2018; Accepted 11/Jan/201
Galerkin approximations for the optimal control of nonlinear delay differential equations
Optimal control problems of nonlinear delay differential equations (DDEs) are
considered for which we propose a general Galerkin approximation scheme built
from Koornwinder polynomials. Error estimates for the resulting
Galerkin-Koornwinder approximations to the optimal control and the value
function, are derived for a broad class of cost functionals and nonlinear DDEs.
The approach is illustrated on a delayed logistic equation set not far away
from its Hopf bifurcation point in the parameter space. In this case, we show
that low-dimensional controls for a standard quadratic cost functional can be
efficiently computed from Galerkin-Koornwinder approximations to reduce at a
nearly optimal cost the oscillation amplitude displayed by the DDE's solution.
Optimal controls computed from the Pontryagin's maximum principle (PMP) and the
Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE
systems, are shown to provide numerical solutions in good agreement. It is
finally argued that the value function computed from the corresponding reduced
HJB equation provides a good approximation of that obtained from the full HJB
equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042
Numerical solution of optimal control problems with constant control delays
We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a state-of-the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.delayed differential equations, delayed optimal control, numerical optimization, time-to-build
On almost sure stability of hybrid stochastic systems with mode-dependent interval delays
This note develops a criterion for almost sure stability of hybrid stochastic systems with mode-dependent interval time delays, which improves an existing result by exploiting the relation between the bounds of the time delays and the generator of the continuous-time Markov chain. The improved result shows that the presence of Markovian switching is quite involved in the stability analysis of delay systems. Numerical examples are given to verify the effectiveness
Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity
Retarded stochastic differential equations (SDEs) constitute a large
collection of systems arising in various real-life applications. Most of the
existing results make crucial use of dissipative conditions. Dealing with "pure
delay" systems in which both the drift and the diffusion coefficients depend
only on the arguments with delays, the existing results become not applicable.
This work uses a variation-of-constants formula to overcome the difficulties
due to the lack of the information at the current time. This paper establishes
existence and uniqueness of stationary distributions for retarded SDEs that
need not satisfy dissipative conditions. The retarded SDEs considered in this
paper also cover SDEs of neutral type and SDEs driven by L\'{e}vy processes
that might not admit finite second moments.Comment: page 2
On input-to-state stability of stochastic retarded systems with Markovian switching
This note develops a Razumikhin-type theorem on pth moment input-to-state stability of hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), which is an improvement of an existing result. An application to hybrid stochastic delay systems verifies the effectiveness of the improved result
Mixed-type functional differential equations: A numerical approach
This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in Journal of computational and applied mathematics and is available at www.elsevier.comThis preprint discusses mixed-type functional equations
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