scholarly journals Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

2020 ◽  
Vol 409 ◽  
pp. 109360 ◽  
Author(s):  
Maurizio Falcone ◽  
Giulio Paolucci ◽  
Silvia Tozza
Keyword(s):  
First Order ◽  
Smoothness Indicators ◽  
Jacobi Equations

scholarly journals An Efficient Filtered Scheme for Some First Order Time-Dependent Hamilton--Jacobi Equations

2016 ◽  
Vol 38 (1) ◽  
pp. A171-A195 ◽  
Author(s):  
Olivier Bokanowski ◽  
Maurizio Falcone ◽  
Smita Sahu
Keyword(s):  
Time Dependent ◽  
First Order ◽  
Jacobi Equations

HOMOGENIZATION OF FIRST-ORDER EQUATIONS WITH u/∊-PERIODIC HAMILTONIAN: RATE OF CONVERGENCE AS ∊ → 0 AND NUMERICAL METHODS

2011 ◽  
Vol 21 (06) ◽  
pp. 1317-1353 ◽  
Author(s):  
YVES ACHDOU ◽  
STEFANIA PATRIZI
Keyword(s):  
Numerical Methods ◽  
Rate Of Convergence ◽  
Effective Hamiltonian ◽  
First Order ◽  
Jacobi Equations ◽  
Homogenization Problems

We consider homogenization problems for first-order Hamilton–Jacobi equations with u∊/∊ periodic dependence, recently introduced by Imbert and Monneau, and also studied by Barles: this unusual dependence leads to nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter ∊ tends to 0. We obtain the same rates as those obtained by Capuzzo Dolcetta and Ishii for the more usual homogenization problems without the dependence in u∊/∊. In the second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.

scholarly journals Regularity for Hamilton-Jacobi equations via approximation

1995 ◽  
Vol 51 (2) ◽  
pp. 195-213 ◽  
Author(s):  
Bum Il Hong
Keyword(s):  
Initial Value Problems ◽  
Stability Theorem ◽  
Polynomial Functions ◽  
Moving Grid ◽  
Approximation Properties ◽  
Initial Value ◽  
First Order ◽  
Spatial Dimensions ◽  
Regularity Results ◽  
Jacobi Equations

We prove new regularity results for solutions of first-order partial differential equations of Hamilton-Jacobi type posed as initial value problems on the real line. We show that certain spaces determined by quasinorms related to the solution's approximation properties in C(ℝ) by continuous, piecewise quadratic polynomial functions are invariant under the action of the differential equation. As a result, we show that solutions of Hamilton-Jacobi equations have enough regularity to be approximated well in C(ℝ) by moving-grid finite element methods. The preceding results depend on a new stability theorem for Hamilton-Jacobi equations in any number of spatial dimensions.

Neighboring Optimal Feedback Law for Higher-Order Dynamic Systems

2002 ◽  
Vol 124 (3) ◽  
pp. 492-497 ◽  
Author(s):  
Tawiwat Veeraklaew ◽  
Sunil K. Agrawal
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Optimal Solution ◽  
Higher Order ◽  
Input State ◽  
Linear Quadratic ◽  
First Order ◽  
Optimal Feedback ◽  
Order Form ◽  
Jacobi Equations

In recent years, using tools from linear and nonlinear systems theory, it has been shown that classes of dynamic systems in first-order forms can be alternatively written in higher-order forms, i.e., as sets of higher-order differential equations. Input-state linearization is one of the most popular tools to achieve such a representation. The equations of motion of mechanical systems naturally have a second-order form, arising from the application of Newton’s laws. In the last five years, effective computational tools have been developed by the authors to compute optimal trajectories of such systems, while exploiting the inherent structure of the dynamic equations. In this paper, we address the question of computing the neighboring optimal for systems in higher-order forms. It must be pointed out that the classical solution of the neighboring optimal problem is well known only for systems in the first-order form. The main contributions of this paper are: (i) derivation of the optimal feedback law for higher-order linear quadratic terminal controller using extended Hamilton-Jacobi equations; (ii) application of the feedback law to compute the neighboring optimal solution.

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