1,720,964 research outputs found
A Multilinear HJB-POD Method for the Optimal Control of PDEs on a Tree Structure
Full text linkOptimal control problems driven by evolutionary partial differential equations arise in many industrial applications and their numerical solution is known to be a challenging problem. One approach to obtain an optimal feedback control is via the Dynamic Programming principle. Nevertheless, despite many theoretical results, this method has been applied only to very special cases since it suffers from the curse of dimensionality. Our goal is to mitigate this crucial obstruction developing a version of dynamic programming algorithms based on a tree structure and exploiting the compact representation of the dynamical systems based on tensors notations via a model reduction approach. Here, we want to show how this algorithm can be constructed for general nonlinear control problems and to illustrate its performances on a number of challenging numerical tests introducing novel pruning strategies that improve the efficacy of the method. Our numerical results indicate a large decrease in memory requirements, as well as computational time, for the proposed problems. Moreover, we prove the convergence of the algorithm and give some hints on its implementation
Optimizing semilinear representations for State-dependent Riccati Equation-based feedback control
Full text linkAn optimized variant of the State Dependent Riccati Equations (SDREs) approach for nonlinear optimal feedback stabilization is presented. The proposed method is based on the construction of equivalent semilinear representations associated to the dynamics and their affine combination. The optimal combination is chosen to minimize the discrepancy between the SDRE control and the optimal feedback law stemming from the solution of the corresponding Hamilton Jacobi Bellman (HJB) equation. Numerical experiments assess effectiveness of the method in terms of stability of the closed-loop with near-to-optimal performance.An optimized variant of the State Dependent Riccati Equations (SDREs) approach for nonlinear optimal feedback stabilization is presented. The proposed method is based on the construction of equivalent semilinear representations associated to the dynamics and their affine combination. The optimal combination is chosen to minimize the discrepancy between the SDRE control and the optimal feedback law stemming from the solution of the corresponding Hamilton Jacobi Bellman (HJB) equation. Numerical experiments assess effectiveness of the method in terms of stability of the closed-loop with near-to-optimal performance
State Dependent Riccati for dynamic boundary control to optimize irrigation in Richards’ equation framework
Full text linkWe present an approach for the optimization of irrigation in a Richards’ equation framework. We introduce a proper cost functional, aimed at minimizing the amount of water provided by irrigation, at the same time maximizing the root water uptake, which is modeled by a sink term in the continuity equation. The control is acting on the boundary of the dynamics and due to the nature of the mathematical problem we use a State Dependent Riccati approach which provides suboptimal control in feedback form, applied to the system of ODEs resulting from the Richards’ equation semidiscretization in space. The problem is tested with existing hydraulic parameters, also considering proper root water uptake functions. The numerical simulations also consider the presence of noise in the model to further validate the use of a feedback control approach
Approximation of Optimal Control Problems for the Navier-Stokes equation via multilinear HJB-POD
Full text linkWe consider the approximation of some optimal control problems for the
Navier-Stokes equation via a Dynamic Programming approach. These control
problems arise in many industrial applications and are very challenging from
the numerical point of view since the semi-discretization of the dynamics
corresponds to an evolutive system of ordinary differential equations in very
high dimension. The typical approach is based on the Pontryagin maximum
principle and leads to a two point boundary value problem. Here we present a
different approach based on the value function and the solution of a Bellman, a
challenging problem in high dimension. We mitigate the curse of dimensionality
via a recent multilinear approximation of the dynamics coupled with a dynamic
programming scheme on a tree structure. We discuss several aspects related to
the implementation of this new approach and we present some numerical examples
to illustrate the results on classical control problems studied in the
literature
FEEDBACK RECONSTRUCTION TECHNIQUES FOR OPTIMAL CONTROL PROBLEMS ON A TREE STRUCTURE
Full text linkThe computation of feedback control using Dynamic Programming equation is a difficult task due the curse of dimensionality. The tree structure algorithm is one the methods introduced recently that mitigate this problem. The method computes the value function avoiding the construction of a space grid and the need for interpolation techniques using a discrete set of controls. However, the computation of the control is strictly linked to control set chosen in the computation of the tree. Here, we extend and complete the method selecting a finer control set in the computation of the feedback. This requires to use an interpolation method for scattered data which allows us to reconstruct the value function for nodes not belonging to the tree. The effectiveness of the method is shown via a numerical example
A HJB-POD approach for the control of nonlinear PDEs on a tree structure
Full text linkThe 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
Error estimates for a tree structure algorithm solving finite horizon control problems
Full text linkIn the Dynamic Programming approach to optimal control problems a crucial
role is played by the value function that is characterized as the unique
viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well
known that this approach suffers of the "curse of dimensionality" and this
limitation has reduced its practical in real world applications. Here we
analyze a dynamic programming algorithm based on a tree structure. The tree is
built by the time discrete dynamics avoiding in this way the use of a fixed
space grid which is the bottleneck for high-dimensional problems, this also
drops the projection on the grid in the approximation of the value function. We
present some error estimates for a first order approximation based on the
tree-structure algorithm. Moreover, we analyze a pruning technique for the tree
to reduce the complexity and minimize the computational effort. Finally, we
present some numerical tests
An efficient DP algorithm on a tree-structure for finite horizon optimal control problems
Full text linkThe 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 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 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
High-order approximation of the finite horizon control problem via a tree structure algorithm
Full text linkSolving optimal control problems via Dynamic Programming is a difficult task that suffers for the”curse of dimensionality”. This limitation has reduced its practical impact in real world applications since the construction of numerical methods for nonlinear PDEs in very high dimension is practically unfeasible. Recently, we proposed a new numerical method to compute the value function avoiding the construction of a space grid and the need for interpolation techniques. The method is based on a tree structure that mimics the continuous dynamics and allows to solve optimal control problems in high-dimension. This property is particularly useful to attack control problems with PDE constraints. We present a new high-order approximation scheme based on the tree structure and show some numerical results
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