Abstract: A new method to approach the problem of solving PDEs by means of Lie series expansion has been deeply investigated by the author (in cooperation with Professor S. Steri) in several papers and the application to biomathematical topics has been performed. The treatment of a cells colony and its evolution has been modelled with a non linear Cauchy problem (evolutionary PDE) and solved by Lie series. The conditions that ensure the existence and the uniqueness of the solution have been presented. Then the control of a drug has been introduced and described by a time dependent parameter.
In our recent studies we are dealing with the following problem: how to introduce the Hamiltonian function in an analytic nonlinear evolutionary process and how to gain a canonical representation of the process by a double series of equations, similarly to what is done in Physics and in finite optimal processes theory. By this way, we are conducted to introduce adjoint variables, namely generalized momenta, to be considered together with the positional variables.
In this work we present a remarkable illustration of the procedure to be followed, developing in details the task for the biological problem above described, the study of a controlled birth and death stochastic process, in which generalized momenta have a clear interpretation. In fact the meaning and the role of new adjoint functions is here suitably discussed. In the end, the possibility to perform an optimum control on the drug action is mentioned, with a particular interest to the application of a minimum principle a la Pontryagin.

Brief Biography of the Speaker:
Prof. Joseph Quartieri is full professor of Physics in the Engineering Faculty of University of Salerno. He belongs to the Physics Department “E.R.Caianiello” of the same University. From 1997-98 he is the coordinator of all the Physics courses in the Engineering Faculty. From 2006 he is also in charge of Medical Physics course at Medicine and Surgery Faculty of University of Salerno.
He got graduated cum laude in Nuclear Physics at Naples University in 1974. From 1980 to 1986 he worked as researcher at National Research Centre (CNR). From 1980 he took several teaching positions as assistant professor, and in 1985 he became associated professor at Engineering Faculty of Rome University “Tor Vergata”. From 1997 he moved to the Engineering Faculty of Salerno University. He got a scientific association with the National Institute for Nuclear Physics (INFN), in the Salerno’s group and participated in the Physics Department “E. R. Caianiello”. He is author of hundreds of papers in several relevant international journals.