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## On an optimal control design for Rossler system

Fonte: Elsevier B.V.
Publicador: Elsevier B.V.

Tipo: Artigo de Revista Científica
Formato: 241-245

ENG

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In this Letter, an optimal control strategy that directs the chaotic motion of the Rossler system to any desired fixed point is proposed. The chaos control problem is then formulated as being an infinite horizon optimal control nonlinear problem that was reduced to a solution of the associated Hamilton-Jacobi-Bellman equation. We obtained its solution among the correspondent Lyapunov functions of the considered dynamical system. (C) 2004 Elsevier B.V All rights reserved.

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## Optimal linear and nonlinear control design for chaotic systems

Fonte: Universidade Estadual Paulista
Publicador: Universidade Estadual Paulista

Tipo: Conferência ou Objeto de Conferência
Formato: 867-873

ENG

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86.34%

#Chaos theory#Computer simulation#Dynamic programming#Feedback control#Hamiltonians#Nonlinear control systems#Optimal control systems#Oscillations#Duffing oscillator#Hamilton Jacobi Bellman equation#Optimal control theory

In this work, the linear and nonlinear feedback control techniques for chaotic systems were been considered. The optimal nonlinear control design problem has been resolved by using Dynamic Programming that reduced this problem to a solution of the Hamilton-Jacobi-Bellman equation. In present work the linear feedback control problem has been reformulated under optimal control theory viewpoint. The formulated Theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations for the Rössler system and the Duffing oscillator are provided to show the effectiveness of this method. Copyright © 2005 by ASME.

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## The patchy Method for the Infinite Horizon Hamilton-Jacobi-Bellman Equation and its Accuracy

Fonte: Escola de Pós-Graduação Naval
Publicador: Escola de Pós-Graduação Naval

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We introduce a modification to the patchy method of Navasca and Krener for solving the stationary Hamilton Jacobi Bellman equation. The numerical solution that we generate is a set of polynomials that approximate the optimal cost and optimal control on a partition of the state space. We derive an error bound for our numerical method under the assumption that the optimal cost is a smooth strict Lyupanov function. The error bound is valid when the number of subsets in the partition is not too large.

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## On the Convergence Rate of the Leake-Liu Algorithm for Solving Hamilton-Jacobi-Bellman Equation

Fonte: International Federation of Automatic Control (IFAC)
Publicador: International Federation of Automatic Control (IFAC)

Tipo: Conference paper

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86.34%

Although many iterative algorithms have been proposed for solving Hamilton- Jacobi-Bellman equation arising from nonlinear optimal control, it remains open how fast those algorithms can converge. The convergence rate of those algorithms is of great import

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## Optimal value of a firm investing in exogeneous technology

Fonte: Instituto Superior de Economia e Gestão
Publicador: Instituto Superior de Economia e Gestão

Tipo: Dissertação de Mestrado

Publicado em //2012
ENG

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#Opções reais de investimento#processos de salto de Poisson#Equações Hamilton-Jacobi-Bellman#Tempos de paragem#Real investment options#Poisson jump processes#Hamilton-Jacobi-Bellman equations#Stopping times

Mestrado em Matemática Financeira; Neste trabalho estudamos o valor ótimo para uma Firma cujo valor função depende de um nível de tecnologia exógeno. Em qualquer ponto no tempo a Firma pode investir numa nova tecnologia incorrendo num custo imediato e em retorno passará a utilizar essa nova tecnologia gerando lucros a partir de uma dada função. Estudamos o tempo de paragem ótimo que corresponde ao ponto no tempo em que a empresa investe para obter a valorização ótima. Usamos uma abordagem de programação dinâmica, encontrando a equação de Hamilton-Jacobi-Bellman cuja solução nos dá o valor ótimo da firma. A
tecnologia é modelada usando um processo estocástico discreto com uma intensidade dependente do tempo. Particularizamos para dois casos, um em que a intensidade é constante e outro em que é dependente do tempo e não monótona.; In this work we study the optimal value for a Firm whose value is function of an exogenous technology level. At any point in time the Firm can invest in a new technology, incurring in an immediate cost and in return it will become able to use that technology yielding profit through a given profit flow function. The technology is modelled by a discrete stochastic process with a time-dependent arrival rate. We study the optimal stopping time that will correspond to the point in time when the firm will invest. We use a dynamic programming approach...

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## Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Quantitative Finance - Portfolio Management#Primary: 35K55, Secondary: 34E05 70H20 91B70 90C15 91B16

In this paper we propose and analyze a method based on the Riccati
transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation
arising from the stochastic dynamic optimal allocation problem. We show how the
fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a
quasi-linear parabolic equation whose diffusion function is obtained as the
value function of certain parametric convex optimization problem. Although the
diffusion function need not be sufficiently smooth, we are able to prove
existence, uniqueness and derive useful bounds of classical H\"older smooth
solutions. We furthermore construct a fully implicit iterative numerical scheme
based on finite volume approximation of the governing equation. A numerical
solution is compared to a semi-explicit traveling wave solution by means of the
convergence ratio of the method. We compute optimal strategies for a portfolio
investment problem motivated by the German DAX 30 Index as an example of
application of the method.

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## Large time behavior of solutions to a degenerate parabolic Hamilton-Jacobi-Bellman equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/09/2015

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We derive the long time asymptotic of solutions to an evolutive
Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with
ergodic problems recently studied in [1]. Our main assumption is an appropriate
degeneracy condition on the operator at the boundary. This condition is related
to the characteristic boundary points for linear operators as well as to the
irrelevant points for the generalized Dirichlet problem, and implies in
particular that no boundary datum has to be imposed. We prove that there exists
a constant c such that the solutions of the evolutive problem converge
uniformly, in the reference frame moving with constant velocity c, to a unique
steady state solving a suitable ergodic problem.

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## Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/02/2009

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This article is devoted to the optimal control of state equations with memory
of the form: ?[x(t) = F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) ds), t>0, with
initial conditions x(0)=x, x(-s)=z(s), s>0.]Denoting by $y_{x,z,u}$ the
solution of the previous Cauchy problem and: \[v(x,z):=\inf_{u\in V}
\{\int_0^{+\infty} e^{-\lambda s} L(y_{x,z,u}(s), u(s))ds \}\] where $V$ is a
class of admissible controls, we prove that $v$ is the only viscosity solution
of an Hamilton-Jacobi-Bellman equation of the form: \[\lambda
v(x,z)+H(x,z,\nabla_x v(x,z))+D_z v(x,z), \dot{z} >=0\] in the sense of the
theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L.
Lions.

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## A Semi-Linear Backward Parabolic cauchy Problem with Unbounded Coefficients of Hamilton-Jacobi-Bellman Type and Applications to optimal control

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/02/2014

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We obtain weighted uniform estimates for the gradient of the solutions to a
class of linear parabolic Cauchy problems with unbounded coefficients. Such
estimates are then used to prove existence and uniqueness of the mild solution
to a semi-linear backward parabolic Cauchy problem, where the differential
equation is the Hamilton-Jacobi-Bellman equation of a suitable optimal control
problem. Via backward stochastic differential equations, we show that the mild
solution is indeed the Value Function of the controlled equation and that the
feedback law is verified.

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## Generalized Hamilton-Jacobi-Bellman equations with Dirichlet boundary and stochastic exit time optimal control problem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We consider a kind of stochastic exit time optimal control problems, in which
the cost function is defined through a nonlinear backward stochastic
differential equation. We study the regularity of the value function for such a
control problem. Then extending Peng's backward semigroup method, we show the
dynamic programming principle. Moreover, we prove that the value function is a
viscosity solution to the following generalized Hamilton-Jacobi-Bellman
equation with Dirichlet boundary: \[ \left\{ \begin{array} [c]{l}
\inf\limits_{v\in V}\left\{\mathcal{L}(x,v)u(x)+f(x,u(x),\nabla u(x)
\sigma(x,v),v)\right\}=0, \quad x\in D,\medskip\\ u(x)=g(x),\quad x\in \partial
D, \end{array} \right. \] where $D$ is a bounded set in $\mathbb{R}^{d}$, $V$
is a compact metric space in $\mathbb{R}^{k}$, and for $u\in C^{2}(D)$ and
$(x,v)\in D\times V$,
\[\mathcal{L}(x,v)u(x):=\frac{1}{2}\sum_{i,j=1}^{d}(\sigma\sigma^{\ast})_{i,j}(x,v)\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}(x) +\sum_{i=1}^{d}b_{i}(x,v)\frac{\partial u}{\partial
x_{i}}(x). \]; Comment: 29 pages

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## Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation
with controlled leading coefficients, which is a type of fully nonlinear
backward stochastic partial differential equation (BSPDE for short). In order
to formulate the weak solution for such kind of BSPDEs, the classical potential
theory is generalized in the backward stochastic framework. The existence and
uniqueness of the weak solution is proved, and for the partially non-Markovian
case, we obtain the associated gradient estimate. As a byproduct, the existence
and uniqueness of solution for a class of degenerate reflected BSPDEs is
discussed as well.; Comment: 29 pages

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## Linear Hamilton Jacobi Bellman Equations in High Dimensions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.; Comment: 8 pages. Accepted to CDC 2014

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## Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We aim to provide a Feynman-Kac type representation for
Hamilton-Jacobi-Bellman equation, in terms of forward backward stochastic
differential equation (FBSDE) with a simulatable forward process. For this
purpose, we introduce a class of BSDE where the jumps component of the solution
is subject to a partial nonpositive constraint. Existence and approximation of
a unique minimal solution is proved by a penalization method under mild
assumptions. We then show how minimal solution to this BSDE class provides a
new probabilistic representation for nonlinear integro-partial differential
equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a
regime switching forward SDE in a Markovian framework, and importantly we do
not make any ellipticity condition. Moreover, we state a dual formula of this
BSDE minimal solution involving equivalent change of probability measures. This
gives in particular an original representation for value functions of
stochastic control problems including controlled diffusion coefficient.; Comment: Published at http://dx.doi.org/10.1214/14-AOP920 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## On traveling wave solutions to Hamilton-Jacobi-Bellman equation with inequality constraints

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Quantitative Finance - Portfolio Management#Mathematics - Analysis of PDEs#35K55, 34E05, 70H20, 91B70, 90C15, 91B16

The aim of this paper is to construct and analyze solutions to a class of
Hamilton-Jacobi-Bellman equations with range bounds on the optimal response
variable. Using the Riccati transformation we derive and analyze a fully
nonlinear parabolic partial differential equation for the optimal response
function. We construct monotone traveling wave solutions and identify
parametric regions for which the traveling wave solution has a positive or
negative wave speed.

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## The patchy Method for the Infinite Horizon Hamilton-Jacobi-Bellman Equation and its Accuracy

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/07/2012

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#Mathematics - Optimization and Control#Mathematics - Numerical Analysis#49-04, 49J15, 49J20, 49L99, 49M37

We introduce a modification to the patchy method of Navasca and Krener for
solving the stationary Hamilton Jacobi Bellman equation. The numerical solution
that we generate is a set of polynomials that approximate the optimal cost and
optimal control on a partition of the state space. We derive an error bound for
our numerical method under the assumption that the optimal cost is a smooth
strict Lyupanov function. The error bound is valid when the number of subsets
in the partition is not too large.; Comment: 50 pages, 5 figures

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## A Reduced Basis Method for the Hamilton-Jacobi-Bellman Equation with Application to the European Union Emission Trading Scheme

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/03/2015

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This paper draws on two sources of motivation: (1) The European Union
Emission Trading Scheme (EU-ETS) aims at limiting the overall emissions of
greenhouse gases. The optimal abatement strategy of companies for the use of
emission permits can be described as the viscosity solution of a
Hamilton-Jacobi-Bellman (HJB) equation. It is a question of general interest,
how regulatory constraints can be set within the EU-ETS in order to reach
certain political goals such as a good balance of emission reduction and
economical growth. Such regulatory constraints can be modeled as parameters
within the HJB equation.
(2) The EU-ETS is just one example where one is interested in solving a
parameterized HJB equation often for different values of the parameters (e.g.\
to optimize their values with respect to a given target functional). The
Reduced Basis Method (RBM) is by now a well-established numerical method to
efficiently solve parameterized partial differential equations. However, to the
best of our knowledge, an RBM for the HJB equation is not known so far and of
(mathematical) interest by its own, since the HJB equation is of hyperbolic
type which is in general a nontrivial task for model reduction.
We analyze and realize a RBM for the HJB equation. In particular...

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## The solution infinite horizon noncooperative differential game with nonlinear dynamics without the Hamilton-Jacobi-Bellman equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

76.45%

For a non-cooperative m-persons differential game, the value functions ofthe
various players satisfy a system of Hamilton-Jacobi-Bellman
equations.Nashequilibrium solutions in feedback form can be obtained by
studying a related system of P.D.E's.A new approach, which is proposed in this
paper allows one to construct the feedback optimal control and cost functions
J_i(t,x),i=1,...,m directly,i.e.,without any reference to the corresponding
Hamilton-Jacobi-Bellman equations.; Comment: 65pages

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## Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

86.38%

The main purpose of this paper is to analyze solutions to a fully nonlinear
parabolic equation arising from the problem of optimal portfolio construction.
We show how the problem of optimal stock to bond proportion in the management
of pension fund portfolio can be formulated in terms of the solution to the
Hamilton-Jacobi-Bellman equation. We analyze the solution from qualitative as
well as quantitative point of view. We construct useful bounds of solution
yielding estimates for the optimal value of the stock to bond proportion in the
portfolio. Furthermore we construct asymptotic expansions of a solution in
terms of a small model parameter. Finally, we perform sensitivity analysis of
the optimal solution with respect to various model parameters and compare
analytical results of this paper with the corresponding known results arising
from time-discrete dynamic stochastic optimization model.; Comment: 20 pages

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## Idempotent/tropical analysis, the Hamilton-Jacobi and Bellman equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/03/2012

Relevância na Pesquisa

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#Mathematics - Rings and Algebras#Mathematical Physics#Mathematics - Functional Analysis#Mathematics - Numerical Analysis#15A80, 35F21, 65F05, 65F99, 65G40, 46E05, 46T99, 52B12

Tropical and idempotent analysis with their relations to the Hamilton-Jacobi
and matrix Bellman equations are discussed. Some dequantization procedures are
important in tropical and idempotent mathematics. In particular, the
Hamilton-Jacobi-Bellman equation is treated as a result of the Maslov
dequantization applied to the Schr\"{o}dinger equation. This leads to a
linearity of the Hamilton-Jacobi-Bellman equation over tropical algebras. The
correspondence principle and the superposition principle of idempotent
mathematics are formulated and examined. The matrix Bellman equation and its
applications to optimization problems on graphs are discussed. Universal
algorithms for numerical algorithms in idempotent mathematics are investigated.
In particular, an idempotent version of interval analysis is briefly discussed.; Comment: 70 pages, 5 figures, CIME lectures (2011), to be published in Lecture
Notes in Mathematics (Springer)

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## Dynamic Programming Principle and Associated Hamilton-Jacobi-Bellman Equation for Stochastic Recursive Control Problem with Non-Lipschitz Aggregator

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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In this work we study the stochastic recursive control problem, in which the
aggregator (or called generator) of the backward stochastic differential
equation describing the running cost is continuous but not necessarily
Lipschitz with respect to the first unknown variable and the control, and
monotonic with respect to the first unknown variable. The dynamic programming
principle and the connection between the value function and the viscosity
solution of the associated Hamilton-Jacobi-Bellman equation are established in
this setting by the generalized comparison theorem of backward stochastic
differential equations and the stability of viscosity solutions. Finally we
take the control problem of continuous-time Epstein-Zin utility with
non-Lipschitz aggregator as an example to demonstrate the application of our
study.

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