The mathematical modelling and numerical simulation of complex systems plays a crucial role in real life applications. Very often, the modelling procedure results in solving a set of partial differential equations (PDEs). However, in many realistic applications, the final goal is not only the mathematical modelling and numerical simulation of the complex system (direct problem), but rather the optimization or optimal control of the considered process (inverse problem).
For instance, in swimming races, the swimmer should optimize his/her stroke while undertaking the least physic al effort; in microelectronics, a given semiconductor device must be designed (for example, by acting on its doping profile) so as to limit power consumption while maximizing some performance index (e.g., the switching speed); in medical surgery, the shape of a by-pass has to be optimized to reduce the vorticity downstream the graft, responsible of a possible new stenosys; in environmental issues, the concentration of toxic substances and pollutants emitted by industrial plants near a town (rather than of contaminants dumped in a river) should be kept below some attention level, without compromising the production rate of the factories involved.
In all these applications, we deal with the optimal control of systems governed by PDEs. The specific objective can actually be formalized in terms of the minimization of a suitable cost functional under the fulfillment of a set of constraints represented by the system of PDEs describing the physical phenomena under investigation.
View a list of journal publications and MOX Reports related to optimal control problems.
View a list of theses related to optimal control problems.