scholarly journals Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

2019 ◽  
Vol 25 ◽  
pp. 15 ◽  
Author(s):  
Manh Khang Dao
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Recent Work ◽  
Comparison Principle ◽  
Value Function ◽  
Exit Costs ◽  
Theory Of Optimal Control ◽  
New Feature ◽  
Jacobi Equations ◽  
The Value Function

We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].

scholarly journals Averaging of the Hamilton-Jacobi equation in infinite dimensions and an application

2000 ◽  
Vol 41 (3) ◽  
pp. 372-385
Author(s):  
Shihong Wang ◽  
Zuoyi Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Viscosity Solution ◽  
Jacobi Equation ◽  
Value Function ◽  
Limit Problem ◽  
Infinite Dimensions ◽  
The Value Function ◽  
Hamilton Jacobi Equation

AbstractWe study the averaging of the Hamilton-Jacobi equation with fast variables in the viscosity solution sense in infinite dimensions. We prove that the viscosity solution of the original equation converges to the viscosity solution of the averaged equation and apply this result to the limit problem of the value function for an optimal control problem with fast variables.

Dynamic programming and feedback analysis of the two dimensional tidal dynamics system

2020 ◽  
Vol 26 ◽  
pp. 109
Author(s):  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Jacobi Equation ◽  
Value Function ◽  
Two Dimensional ◽  
Tidal Dynamics ◽  
Infinite Dimensional ◽  
Bellman Principle ◽  
The Value Function ◽  
Hamilton Jacobi Equation

In this work, we consider the controlled two dimensional tidal dynamics equations in bounded domains. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled 2D tidal dynamics equations. The existence of an optimal control is shown and the dynamic programming method for the optimal control of 2D tidal dynamics system is also described. We show that the feedback control can be obtained from the solution of an infinite dimensional Hamilton-Jacobi equation. The non-differentiability and lack of smoothness of the value function forced us to use the method of viscosity solutions to obtain a solution of the infinite dimensional Hamilton-Jacobi equation. The Bellman principle of optimality for the value function is also obtained. We show that a viscosity solution to the Hamilton-Jacobi equation can be used to derive the Pontryagin maximum principle, which give us the first order necessary conditions of optimality. Finally, we characterize the optimal control using the adjoint variable.

scholarly journals Basel III and the Net Stable Funding Ratio

2013 ◽  
Vol 2013 ◽  
pp. 1-20 ◽  
Author(s):  
F. Gideon ◽  
Mark A. Petersen ◽  
Janine Mukuddem-Petersen ◽  
LNP Hlatshwayo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Continuous Time ◽  
Value Function ◽  
Analytic Solution ◽  
Market Liquidity ◽  
Basel Iii ◽  
Quantitative Manner ◽  
The Value Function

We validate the new Basel liquidity standards as encapsulated by the net stable funding ratio in a quantitative manner. In this regard, we consider the dynamics of inverse net stable funding ratio as a measure to quantify the bank’s prospects for a stable funding over a period of a year. In essence, this justifies how Basel III liquidity standards can be effectively implemented in mitigating liquidity problems. We also discuss various classes of available stable funding and required stable funding. Furthermore, we discuss an optimal control problem for a continuous-time inverse net stable funding ratio. In particular, we make optimal choices for the inverse net stable funding targets in order to formulate its cost. This is normally done by obtaining analytic solution of the value function. Finally, we provide a numerical example for the dynamics of the inverse net stable funding ratio to identify trends in which banks behavior convey forward looking information on long-term market liquidity developments.

scholarly journals On the Bellman equations with varying control

1996 ◽  
Vol 53 (1) ◽  
pp. 51-62 ◽  
Author(s):  
Shigeaki Koike
Keyword(s):  
Viscosity Solution ◽  
Viscosity Solutions ◽  
Comparison Principle ◽  
Bellman Equation ◽  
Value Function ◽  
Cost Functional ◽  
Bellman Equations ◽  
First Order ◽  
The Value Function ◽  
Admissible Controls

The value function is presented by minimisation of a cost functional over admissible controls. The associated first order Bellman equations with varying control are treated. It turns out that the value function is a viscosity solution of the Bellman equation and the comparison principle holds, which is an essential tool in obtaining the uniqueness of the viscosity solutions.

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