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In this article, we show that under reasonable assumptions every Lipschitz-continuous solution to a Hamilton–Jacobi inequality approximates witha prioriknown error the optimal value of a respective Bolza functional and that such approximation is stable. The solutions of Hamilton–Jacobi variational inequalities can be easily obtained by well-known numerical methods as approximate solutions of Hamilton–Jacobi equations resulting from related Bolza functionals. The main strength of this approach lies in the fact that both precise solution to the Hamilton–Jacobi PDE and the distance between that solution and its numerical approximation need not be known in order to solve the original Bolza problem. [ABSTRACT FROM PUBLISHER]

In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed. [ABSTRACT FROM AUTHOR]