Controlled diffusion processes

Borkar, Vivek S.

doi:10.1214/154957805100000131

Cited by 64 publications

(36 citation statements)

References 125 publications

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“…This is a Bolza's type cost functional in the finite-horizon case (T < ∞) and it is supposed that the controller knows the state of the system at each instant of time (complete observations). For this case, the method of dynamic programming can be applied [2,4] in order to write the Hamilton-Jacobi-Bellman (HJB) equation for inf u J with u as optimization function. Some other cases of the cost structure of J(X, u) are quoted in [2], that have application in finance, engineering, and in production planning and forest harvesting.…”

Section: Introductionmentioning

confidence: 99%

“…For this case, the method of dynamic programming can be applied [2,4] in order to write the Hamilton-Jacobi-Bellman (HJB) equation for inf u J with u as optimization function. Some other cases of the cost structure of J(X, u) are quoted in [2], that have application in finance, engineering, and in production planning and forest harvesting. Each J will lead to a different form of the HJB equation that can be analyzed with appropriate methods of partial differential equations [3].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Control of Probability Density Functions of Stochastic Processes

Annunziato

Borzì

2010

Mathematical Modelling and Analysis

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Abstract. A Fokker-Planck framework for the formulation of an optimal control strategy of stochastic processes is presented. Within this strategy, the control objectives are defined based on the probability density functions of the stochastic processes. The optimal control is obtained as the minimizer of the objective under the constraint given by the Fokker-Planck model. Representative stochastic processes are considered with different control laws and with the purpose of attaining a final target configuration or tracking a desired trajectory. In this latter case, a receding-horizon algorithm over a sequence of time windows is implemented.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal Control of Probability Density Functions of Stochastic Processes

Annunziato

Borzì

2010

Mathematical Modelling and Analysis

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“…2 It is known that under this setting the SDE (1) admits a unique strong solution [Bor05]. We let (X t,x;u s ) s≥t denote the unique strong solution of (1) starting from time t at the state x under the control u.…”

Section: The Setting and Statement Of Problemmentioning

confidence: 99%

The stochastic reach-avoid problem and set characterization for diffusions

2016

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Abstract. In this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective. Preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost-sure stochastic requirements. In contrast, we propose a methodology to tackle problems with less stringent requirements than almost sure. To this end, we first establish a connection between two distinct stochastic reach-avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions. Subsequently, we focus on solutions of one of the classes of stochastic optimal control problems-the exit-time problem, which solves both the two reach-avoid problems mentioned above. We then derive a weak version of a dynamic programming principle (DPP) for the corresponding value function; in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions. Moreover, based on our DPP, we provide an alternative characterization of the value function as a solution of a partial differential equation in the sense of discontinuous viscosity solutions, along with boundary conditions both in Dirichlet and viscosity senses. Theoretical justifications are also discussed to pave the way for deployment of off-the-shelf PDE solvers for numerical computations. Finally, we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.

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“…To this end, we need to topologize M i suitably. Here we use the L 2 weak * -topology on M i , i = 1, 2, as in [3,Chapter 2]. The space endowed with this topology is compact and metrizable.…”

Section: Average Payoffmentioning

confidence: 99%

“…The space endowed with this topology is compact and metrizable. Further, the topology on M i can be characterized as follows, see [3,Chapter 2 ].…”

Section: Average Payoffmentioning

confidence: 99%