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

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

A tree structure algorithm for optimal control problems with state constraints

We present a tree structure algorithm for optimal control problems with state constraints. We prove a convergence result for a discrete time approximation of the value function based on a novel formulation in the case of convex constraints. Then the Dynamic Programming approach is developed by a discretization in time leading to a tree structure in space derived by the controlled dynamics, taking into account the state constraints to cut several branches of the tree. Moreover, an additional pruning allows for the reduction of the tree complexity as for the case without state constraints. Since the method does not use an a priori space grid, no interpolation is needed for the reconstruction of the value function and the accuracy essentially relies on the time step h. These features permit a reduction in CPU time and in memory allocations. The synthesis of optimal feedback controls is based on the values on the tree and an interpolation on the values obtained on the tree will be necessary if a different discretization in the control space is adopted, e.g. to improve the accuracy of the method in the reconstruction of the optimal trajectories. Several examples show how this algorithm can be applied to problems in low dimension and compare it to a classical DP method on a grid

Aptitud forrajera de líneas S2 originadas del híbrido Zea mays L. x Zea diploperennis I.

Nineteen Iines of forage com originated from the crossing among Zea mays x Zea diploperennis in the second year of self-fertilization were evaluated. The analyzed characteristics were: plant height, number of stems per plant ''. number of ears per plant', stem diameter and leaf I stem relationship. Rainfall and temperature conditions were optimum during the com cultivation cycle. It was found difference among Iines in all the evaluated characteristics. Lines 3 and 16 exceeded the average for all the forage attributes. Lines 5 and 15 w1thsimilar aptitud do not overcome leaf/stem relationship. The general combinity aptitude will be able to identify the lines with the upper value of digestible dry matter.Se analizaron 19 líneas de maíz forrajero originadas a partir del cruzamiento entre Zea mays L. x Zea diploperennis l. en su segundo año de autofecundación (S2). Las características analizadas fueron: altura de planta, número de tallos por planta, número de mazorcas por planta, diámetro del tallo y la relación hoja/tallo. Las condiciones de precipitaciones y temperaturas fueron óptimas durante el ciclo del cultivo de maíz. Se encontró diferencia entre las líneas en todas las características evaluadas. Las líneas 3 y 16 superaron el promedio en todos los atributos forrajeros. Por su parte, la 5 y 15 también lo hicieron, aunque no superaron el promedio en la relación hoja/tallo. La evaluación de la Aptitud Combinatoria General permitiré identificar las líneas que maximicen la producción de materia seca digestible

FEEDBACK RECONSTRUCTION TECHNIQUES FOR OPTIMAL CONTROL PROBLEMS ON A TREE STRUCTURE

The 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

An efficient DP algorithm on a tree-structure for finite horizon optimal control problems

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 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