Suboptimal Feedback Control of PDEs by Solving HJB Equations on Adaptive Sparse Grids

Garcke, Jochen; Kröner, Axel

doi:10.1007/s10915-016-0240-7

Cited by 62 publications

(52 citation statements)

References 45 publications

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“…We refer, among others, to the pioneer work on the coupling between model reduction and HJB approach [24] and the recent [3] work which provides a-priori error estimates for the aforementioned coupling method. We also mention a sparse grid approach in [18] where the authors apply HJB to the control of the wave equation and a spectral elements approximation in [23] which allows to solve the HJB equation up to dimension 12. Despite these efforts and the mathematical elegance of the DP approach, its impact in industrial applications is limited by this bottleneck and the solution of many optimal control problems has been accomplished instead via open-loop control.…”

mentioning

confidence: 99%

An Efficient DP Algorithm on a Tree-Structure for Finite Horizon Optimal Control Problems

Alla¹,

Falcone²,

Saluzzi³

2019

SIAM J. Sci. Comput.

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The classical Dynamic Programming (DP) approach to optimal control problems is based on the characterization of the value function as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. The DP scheme for the numerical approximation of viscosity solutions of Bellman equations is typically based on a time discretization which is projected on a fixed state-space grid. The time discretization can be done by a one-step scheme for the dynamics and the projection on the grid typically uses a local interpolation. Clearly the use of a grid is a limitation with respect to possible applications in high-dimensional problems due to the curse of dimensionality.Here, we present a new approach for finite horizon optimal control problems where the value function is computed using a DP algorithm with a tree structure algorithm (TSA) constructed by the time discrete dynamics. In this way there is no need to build a fixed space triangulation and to project on it. The tree will guarantee a perfect matching with the discrete dynamics and drop off the cost of the space interpolation allowing for the solution of very high-dimensional problems. Numerical tests will show the effectiveness of the proposed method.

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mentioning

confidence: 99%

An Efficient DP Algorithm on a Tree-Structure for Finite Horizon Optimal Control Problems

Alla¹,

Falcone²,

Saluzzi³

2019

SIAM J. Sci. Comput.

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“…and, the POD approximation for (4) reads as follows: min u∈U J x ,t (u) such that y (t) solves (16).…”

Section: Pod For the Optimal Control Problemmentioning

confidence: 99%

A HJB-POD approach for the control of nonlinear PDEs on a tree structure

Alla

Saluzzi

2020

Applied Numerical Mathematics

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The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation.Here, we aim at testing the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.

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“…[32] for a polynomial approximation of high-dimensional HJB equations. In the context of suboptimal control of PDEs an approach to solve the associated HJB equation based on sparse grids was considered in [23]. Here, we use a finite difference scheme method; more precisely, an essentially non-oscillatory (ENO) scheme [48] for space discretization is coupled with a Runge-Kutta time discretization scheme of second order following [7,48].…”

Section: 34mentioning

confidence: 99%

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

Chekroun¹,

Kröner²,

Liu³

2018

Hamilton-Jacobi-Bellman Equations

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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. 26 References 26

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Suboptimal Feedback Control of PDEs by Solving HJB Equations on Adaptive Sparse Grids

Cited by 62 publications

References 45 publications

An Efficient DP Algorithm on a Tree-Structure for Finite Horizon Optimal Control Problems

An Efficient DP Algorithm on a Tree-Structure for Finite Horizon Optimal Control Problems

A HJB-POD approach for the control of nonlinear PDEs on a tree structure

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

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