On the gradient in optimization problems of nonstationary systems with distributed control

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 ∇J(u;x,t) --> ∇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.