Abstract:
For the first time, the problem of determining the gradient, not the Fréchet derivative, of a functional $J(u)$ for numerical optimization problems with non stationary partial differential systems under control $u(x)$ is discussed. It is shown that control should be considered as a function of both space $x$ and time $t$. The controllability of such a task is investigated, taking into account the mapping of the space-time gradient $\nabla J(u;x,t) \to \nabla J(u;x)$ by traditional time integration and projection onto the line $x$ at the right moment $t$. Examples are considered: identification of the roughness of an open channel, optimal design of the nozzle shape of a hydraulic gun. It is revealed that optimization with a new gradient form on the line implements the best approximation to the optimum. When optimizing the nozzle shape, new optimal shapes were found.
Keywords:gradient, optimization, controllability, open channel, nozzle.
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