OPTIMAL CONTROL OF A PHASE FIELD TUMOR GROWTH MODEL WITH CHEMOTAXIS AND ACTIVE TRANSPORT

This paper is concerned with a distributed optimal control problem for a phase field model describing tumor growth with chemotaxis and active transport. First, comparing with the results in [H. Garcke, K.F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport, European J. Appl. Math. 28 (2017), 284-316], we prove the existence of solutions for such a system with more general potential, the regularity of solutions and the continuous dependence of initial data as well as control variable with respect to a strong topology. It is worth pointing out that the potentials cover the case of classical quartic double-well potential, which is the standard approximation for the physical relevant logarithmic potential. Furthermore, the existence of an optimal control is proved by monotonicity arguments and compactness theorems. Beyond that, by overcoming some difficulties in mathematical analysis and calculation, especially in the proof of the Fre ' chet differentiability of the control-to-state operator, we derive the corresponding first-order necessary conditions of optimality in terms of the adjoint variables and the usual variational inequality.