Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control

Controlling conservation laws I: entropy-entropy flux

We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux--gradient flows". We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers' equation, are presented. Incorporating both primal-dual algorithms and monotone schemes, we successfully com…

Symbolic Computation in the Calculus of Variations: Determination of Symmetries and Conservation Laws (in Portuguese)

English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking, nonlinear, and very hard to solve. One way to address the problem is to obtain conservation laws of lower order than those of the corresponding Euler-Lagrange equations. While in Physics and Economics the question of existence of conservation laws is treated in a rather natural way, because the application itself suggest the conservation laws (e.g., conservation of e…

非局部守恒律的最优控制问题：优化解的存在性及局部守恒律近似中的奇异性极限

Optimal Control Problems with Nonlocal Conservation Laws: Existence of Optimizers and Singular Limits in Approximations of Local Conservation Laws

本文研究非局部守恒律下的最优控制问题，证明了优化解的存在性，并展示了当非局部核趋近于狄拉克分布时，解向局部守恒律问题的收敛性，同时提供了数值验证。

Weak calculus of variations for functionals of laws of semi-martingales

We develop a non-anticipating calculus of variations for functionals on a space of laws of continuous semi-martingales, which extends the classical one. We extend Hamilton's least action principle and Noether's theorem to this generalized stochastic framework. As an application we obtain, under mild conditions, a stochastic Euler-Lagrange condition and invariants for the critical points of recent problems in stochastic control, namely for the semi-martingale optimal transportation problems.

On the Boundary Control of Systems of Conservation Laws

The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general.

Quasi-potentials of the entropy functionals for scalar conservation laws

We investigate the quasi-potential problem for the entropy cost functionals of non-entropic solutions to scalar conservation laws with smooth fluxes. We prove that the quasi-potentials coincide with the integral of a suitable Einstein entropy.