Weighted functional spaces approach to non-convex nonlinear infinite horizon optimal control problems: stabilization, numerical analysis, bio-medical applications.

• Applicants: Dr. Valeriya Lykina; Professorin Dr. Sabine Pickenhain
• Subject Area: Mathematics ; Epidemiology and Medical Biometry/Statistics
• Term: since 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 387882680

In the project biomedical models are formulated and treated as infinite horizon optimal control problems. Among them one considers two classes of problems: infinite horizon optimal control problems of 1) the combined chemotherapeutic-antiangiogenic cancer treatment; 2) the non-pharmaceutical epidemiological intervention with non-convex isolation costs. The incorporation of an unbounded planing horizon for the considered models has been investigated in our previous projects and represents an important and challenging mathematical issue which has been proved to be a proper idealization for sustainability principle so much desired and targeted by the today's society. One important aspect leading to an infinite horizon formulation is the fact that it is not reasonable to assume to know the end point of e.g. epidemics or the decease, cf. spread of COVID-19 or cancer. Another important aspect results from the point of view of stabilization the controlled dynamic system, since the decision maker try to stabilize the number of infected around some specific level. Both optimization and stabilization goals are properly to combine in the infinite horizon objective functional. The formulated models, besides the direct interest in the long-term optimal strategies in the particular background, should serve as benchmark problems to detect and illustrate the observed mathematical phenomena as well. Fundamental research goals of the project cover both theoretical aspects of solving nonconvex infinite horizon optimal control problems by means of relaxation techniques and development of suitable numerical solution methods for the mentioned class of problems which is strongly required due to nonlinearity of the problems. The first part is based on relaxation ideas of Gamkrelidze applied to the considered problem class as well on the duality concept of Klötzler adjusted to the setting of the OCP in weighted functional spaces. Derivation and proof of sufficient optimality conditions and Pontryagin Type Maximum Principle is the main theoretical goal of the project. The second part includes the development of a pseudospectral method for numerical solution of the primal as well as the dual problems which combines the concepts of duality theory and the relaxation techniques. The convergence proofs and the comparison to the previously developed dual-based direct pseudo-spectral method are to establish. Especially the solving of stabilization problems by means of relaxed infinite horizon optimal control problems in the context of functional analytical approach represents a new and challenging research area. From practical point of view it is particularly interesting to detect the so called "dithering" solutions for problems with nonconvex objectives and understand their meaning and applicability in case of biomedical applications.

• DFG Programme: Research Grants