engleski

Distributed optimal control of parabolic equations by spectral decomposition

We consider the constrained minimisation problem ( P ) min u ∈ L T , U 2 {; ; J ( u ) : x ( T ) ∈ B ε ( x T ) ― }; ; , where x T is some given target state, J is a given cost functional and x is the solution of ( E ) {; ; d d t x(t)+ A x(t)= B t u(t) for t∈ ( 0 , T ) x(0)=0. Here A is an unbounded linear operator allowing for spectral decomposition, while B t is a (time dependent) control operator. If the cost functional J is given by J ( u ) = ‖ u ‖ L 2 the problem ( P ) is reduced to a classical minimal norm control problem which can be solved by Hilbert uniqueness method (HUM). In [1] we allow for a more general cost functional and analyse examples in which, apart from the target state and the control norm, one considers a desired trajectory and penalise a distance of the state from it. Such problem requires a more general approach, and it has been addressed by different methods throughout last decades. In this paper we suggest another method based on the spectral decomposition in terms of eigenfunctions of the operator A . Surprisingly, the problem reduces to a non-linear equation for a scalar unknown, representing a Lagrangian multiplier. The same approach has been recently introduced in [2] for an optimal control problem of the heat equation in which the control was given through the initial datum. This paper generalises the method to the distributed control problems. As can be expected, in this case one has to consider the associated dual problem which makes the calculation more complicated, although the algorithm steps follow a similar structure as in [2]. In the talk basic steps of the method will be explained, followed by numerical examples demonstrating its efficiency.

convex optimization ; dual problem ; , optimal control ; parabolic equations ; spectral decomposition

nije evidentirano