# Approximate control of parabolic equations by spectral decomposition

Friday, June 16th, 2017, 12:00h.,
Central Meeting Room at DeustoTech

#### Martin Lazar

University of Dubrovnik, Dubrovnik, Croatia

#### Cesare Molinari

Universidad Técnica Federico Santa María, Valparaíso, Chile.

Abstract:
We consider the constrained minimisation problem

where $x^T$ is some given target state, J is a given cost functional and x is the solution of

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 this paper 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 dierent 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 an algebraic equation for a scalar unknown, representing a Lagrangian multiplier. The same approach has been recently introduced in [1] 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 [1]. In the talk basic steps of the method will be explained, followed by numerical examples demonstrating its efficiency.

###### References

[1] Lazar, M, Molinari C, J. Peypouquet Optimal control by spectral decomposition of parabolic equations , Optimisation, (2017)