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## On an optimal control design for Rossler system

Rafikov, M.; Balthazar, José Manoel
Fonte: Elsevier B.V. Publicador: Elsevier B.V.
Tipo: Artigo de Revista Científica Formato: 241-245
ENG
Relevância na Pesquisa
86.34%
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.

## Optimal linear and nonlinear control design for chaotic systems

Rafikov, Marat; Balthazar, José Manoel
Tipo: Conferência ou Objeto de Conferência Formato: 867-873
ENG
Relevância na Pesquisa
86.34%
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.

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

Hunt, Thomas; Krener, Arthur J.
Relevância na Pesquisa
86.34%
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.

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

Chen, Weitian; Anderson, Brian
Fonte: International Federation of Automatic Control (IFAC) Publicador: International Federation of Automatic Control (IFAC)
Tipo: Conference paper
Relevância na Pesquisa
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

## Optimal value of a firm investing in exogeneous technology

Pólvora, Pedro Ribeiro Coelho Fouto
Fonte: Instituto Superior de Economia e Gestão Publicador: Instituto Superior de Economia e Gestão
Relevância na Pesquisa
76.55%
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...

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

Kilianova, Sona; Sevcovic, Daniel
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
86.57%
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.

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

Castorina, Daniele; Cesaroni, Annalisa; Rossi, Luca
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
86.34%
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.

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

Carlier, Guillaume; Tahraoui, Rabah
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.34%
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.

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

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.38%
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.

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

Buckdahn, Rainer; Nie, Tianyang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.34%
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

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

Qiu, Jinniao
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.34%
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

## Linear Hamilton Jacobi Bellman Equations in High Dimensions

Horowitz, Matanya B.; Damle, Anil; Burdick, Joel W.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.34%
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

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

Kharroubi, Idris; Pham, Huyên
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.51%
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)

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

Ishimura, Naoyuki; Sevcovic, Daniel
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.34%
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.

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

Hunt, Thomas; Krener, Arthur J.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
86.34%
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

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

Steck, Sebastian; Urban, Karsten
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
76.42%
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...

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

Foukzon, Jaykov
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

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

Macova, Zuzana; Sevcovic, Daniel
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

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

Litvinov, Grigory L.
Tipo: Artigo de Revista Científica