scholarly journals Numerical Methods for the Optimal Control of Scalar Conservation Laws

2013 ◽  
pp. 136-144 ◽  
Author(s):  
Sonja Steffensen ◽  
Michael Herty ◽  
Lorenzo Pareschi
Keyword(s):  
Optimal Control ◽  
Numerical Methods ◽  
Conservation Laws ◽  
Scalar Conservation Laws ◽  
Scalar Conservation

scholarly journals Optimal controllability for scalar conservation laws with convex flux

2014 ◽  
Vol 11 (03) ◽  
pp. 477-491 ◽  
Author(s):  
Adimurthi ◽  
Shyam Sundar Ghoshal ◽  
G. D. Veerappa Gowda
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Conservation Laws ◽  
Linear Problem ◽  
Optimal Solution ◽  
Scalar Conservation Laws ◽  
Existence Of A Solution ◽  
New Strategy ◽  
Scalar Conservation

The optimal control problem for Burgers equation was first considered by Castro, Palacios and Zuazua. They proved the existence of a solution and proposed a numerical scheme to capture an optimal solution via the method of "alternate decent direction". In this paper, we introduce a new strategy for the optimal control problem for scalar conservation laws with convex flux. We propose a new cost function and by the Lax–Oleinik explicit formula for entropy solutions, the nonlinear problem is converted to a linear problem. Exploiting this property, we prove the existence of an optimal solution and, by a backward construction, we give an algorithm to capture an optimal solution.

scholarly journals Continuous dependence and error estimation for viscosity methods

Acta Numerica ◽  
2003 ◽  
Vol 12 ◽  
pp. 127-180 ◽  
Author(s):  
Bernardo Cockburn
Keyword(s):  
Numerical Methods ◽  
Conservation Laws ◽  
Approximation Theory ◽  
Continuous Dependence ◽  
Entropy Solution ◽  
Viscosity Coefficient ◽  
Scalar Conservation Laws ◽  
Natural Approach ◽  
Scalar Conservation ◽  
Continuous Dependence Results

In this paper, we review some ideas on continuous dependence results for the entropy solution of hyperbolic scalar conservation laws. They lead to a complete L^\infty(L^1)-approximation theory with which error estimates for numerical methods for this type of equation can be obtained. The approach we consider consists in obtaining continuous dependence results for the solutions of parabolic conservation laws and deducing from them the corresponding results for the entropy solution. This is a natural approach, as the entropy solution is nothing but the limit of solutions of parabolic scalar conservation laws as the viscosity coefficient goes to zero.

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