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Mathematical Analysis

To find optimal solutions to our control problem (11)-(13) we make use of Pontryagin's Maximum Principle and formulate the current value Hamiltonian

 where  and  denote the current value adjoint variables to  and , respectively. Note that we set  for further convenience.

As the optimal change  of the blood sucking rate  has to maximize the Hamiltonian, the assumptions made on the function  imply that  has to follow

 and hence .

The time derivatives of the adjoint variables are given by


As the Hamiltonian (14) is concave jointly in the states and the control, the canonical system consisting of the two state equations (12) and (13), the adjoint equations (16) and (17) and the optimality condition (15) together with the limiting transversality condition


are not only necessary but even sufficient conditions for optimal solutions of our problem. Our aim is to show that the canonical system possesses stable periodic solutions. One way to do this is to apply the Hopf bifurcation theorem. Among other things this requires that the canonical system has an equilibrium and that the Jacobian evaluated at the steady state possesses a pair of purely imaginary eigenvalues and no other eigenvalue with zero real part.

First we want to establish the existence of a steady state. A steady state  of the canonical system is a solution to the following system of nonlinear equations:


From (20) and (23)  and  follow immediately. Moreover, (19) and (22) yield the relations


From (24) and (21) it can be seen that the steady state value  has to be a solution of


Specifying  as a linear function  with  is monotonically increasing and therefore at most one steady state can exist. Moreover, if


hold, one and only one equilibrium exists.

    The Transylvanian problem
    Mathematical Analysis
    Stability Analysis