Many real life problems are described by mathematical models depending on a number of parameters representing some variable physical coefficients, e.g. temperature of a conductive material, relative humidity of air, turbulence friction coefficient etc. However, in practical applications, the models under consideration are often not completely known, submitted to unknown or uncertain parameters, either of deterministic or of stochastic nature. It is therefore essential to develop robust analytical and computational methods, not only allowing to control a given model, but also to deal with parameter-dependent families of systems in a stable and computationally efficient way.
As a first step in control of parameter dependent systems, the notion of averaged control was introduced recently. Its goal is to control the average of parameter-dependent system components by a single control. The positive results have been obtained both for finite and infinite dimensional systems, as well as for deterministic and stochastic parameter dependence.
Greedy control represents another approach in that direction, based on the complementary issue of determining the most relevant values of the unknown parameters so in order to provide the best possible approximations of all parameter-depending controls. The analysis is based on previous work on reduced modelling and (weak) greedy algorithms for solving parameter-depending partial differential equations (PDEs) or abstract equations in Banach spaces. The obtained results cover the exact controllability of systems ordinary differential equations (ODEs), either of finite or infinite dimension, governed by a bounded operator.
Within the proposed research we want to extend the greedy control results toward several directions: develop greedy control approach for the approximate controllability problem for finite dimensional systems, as well as for the systems governed by unbounded operators and PDEs, both on the exact and the approximate control level.
Literature:
- DeVore, Ron.
The Theoretical Foundation of Reduced Basis Methods. To appear in Model Reduction and Approximation - Lazar, Martin; Zuazua, Enrique.
Greedy controllability of finite dimensional linear systems. Automatica. 74 (2016) ; 327-340 - Hernández Santamaría, Víctor; Lazar, Martin; Zuazua, Enrique.
Greedy optimal control for elliptic control problems and its application to turnpike problems, submitted