This article proposes a new approach for computing a semi-explicit form of the solution to a class of Hamilton-Jacobi (HJ) partial differential equations (PDEs), using control techniques based on viability theory. We characterize the epigraph of the value function solving the HJ PDE as a capture basin of a target through an auxiliary dynamical system, called ?characteristic system?. The properties of capture basins enable us to define components as building blocks of the solution to the HJ PDE in the Barron/Jensen-Frankowska sense. These components can encode initial conditions, boundary conditions, and internal ?boundary? conditions, which are the topic of this article. A generalized Lax-Hopf formula is derived, and enables us to formulate the necessary and sufficient conditions for a mixed initial and boundary conditions problem with multiple internal boundary conditions to be well posed. We illustrate the capabilities of the method with a data assimilation problem for reconstruction of highway traffic flow using Lagrangian measurements generated from Next Generation Simulation (NGSIM) traffic data.

Lax–Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part I: Theory

We use viability techniques for solving Dirichlet problems with inequality constraints (obstacles) for a class of Hamilton-Jacobi equations. The hypograph of the “solution” is defined as the “capture basin” under an auxiliary control system of a target associated with the initial and boundary conditions, viable in an environment associated with the inequality constraint. From the tangential condition characterizing capture basins, we prove that this solution is the unique “upper semicontinuous” solution to the Hamilton-Jacobi-Bellman partial differential equation in the Barron-Jensen/Frankowska sense. We show how this framework allows us to translate properties of capture basins into corresponding properties of the solutions to this problem. For instance, this approach provides a representation formula of the solution which boils down to the Lax-Hopf formula in the absence of constraints.

/pdf/dirichlet-problems-for-some-hamilton-jacobi-equations-with-wdbhj43mpz.pdf

Dirichlet Problems for some Hamilton-Jacobi Equations with Inequality Constraints

Consider the minimal time control problem for a single-input control-affine system x = X(x) + u1Y1(x) in Rn, where the scalar control u1(·) satisfies the constraint lu1(·)l ≤. When applying a shooting method for solving this kind of optimal control problem, one may encounter numerical problems due to the fact that the shooting function is not smooth whenever the control is bang-bang. In this article we propose the following smoothing procedure. For e > 0 small, we consider the minimal time problem for the control system x = X(x) + u1eY1(x) + eΣi=2m uieYi(x), where the scalar controls uie(·), i = 1,... , m, with m ≥ 2, satisfy the constraint Σi=1m uie(uie(t))2 ≤ 1. We prove, under appropriate assumptions, a strong convergence result of the solution of the regularized problem to the solution of the initial problem.

https://hal.archives-ouvertes.fr/hal-00414680/document

Smooth Regularization of Bang-Bang Optimal Control Problems

We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and nonlocal eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows us to prove that the energy associated to the dislocation dynamics is nonincreasing.

https://www.ems-ph.org/journals/show_pdf.php?issn=1463-9963&vol=7&iss=4&rank=3

Existence and uniqueness for dislocation dynamics with nonnegative velocity

This conference paper is a summary of the article “Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints”, J.-P. Aubin, A. Bayen, P. Saint-Pierre, SIAM Journal on Control and Optimization, 47(5), pp. 23482380, 2008, doi:10.1137/060659569. The full article contains all proofs and theorems summarized here. We use viability techniques for solving Dirichlet problems with inequality constraints (obstacles) for a class of Hamilton-Jacobi equations. The hypograph of the “solution” is defined as the “capture basin” under an auxiliary control system of a target associated with the initial and boundary conditions, viable in an environment associated with the inequality constraint. From the tangential condition characterizing capture basins, we prove that this solution is the unique “upper semicontinuous” solution to the Hamilton-Jacobi-Bellman partial differential equation in the Barron-Jensen/Frankowska sense. We show how this framework allows us to translate properties of capture basins into corresponding properties of the solutions to this problem. For instance, this approach provides a representation formula of the solution which boils down to the Lax-Hopf formula in the absence of constraints.

/pdf/dirichlet-problems-for-some-hamilton-jacobi-equations-with-14wzgdh3r8.pdf

Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints

In this paper the authors use the method of characteristics to extend the Jacobi conjugate points theory to the Bolza problem arising in nonlinear optimal control. This yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation. The same approach allows to investigate as well smoothness of the value function.

/pdf/conjugate-points-and-shocks-in-nonlinear-optimal-control-2leauqeg71.pdf

N. Caroff

Papers

Conjugate points and shocks in nonlinear optimal control