Nonconcavity and proper optimal periodic control (Q5906562)

It has been well-known since the sixties that optimal control problems in which the differential equation describing the model and the criterion function do not explicitly depend on time may allow periodic solutions that give rise to larger (i.e. better) values of the criterion function than constant solutions. If such periodic solutions exist, one calls the OPC (optimal periodic control) problem proper. Necessary conditions as well as sufficiency conditions exist for an OPC to be proper. The current paper starts with the problem statement and a brief description of such already known conditions. No necessary and sufficient conditions are known at this time. The present paper gives a new necessary condition for general OPC problems to be proper. The most essential part of this necessary condition is that the Hamiltonian (at it appears in the maximum principle of Pontryagin) be nonconcave in the co-state variable \(\lambda\). Two cases can be distinguished here: the period \(T\) is fixed or it is not and in the latter case \(T\) should be chosen optimally. In the last part of the paper the theory developed is applied to an economic example in which the model is a two-dimensional differential equation describing how unemployment and inflation might interact. A criterion function is given as well and the resulting Hamiltonian is not concave. Analytical solutions are not possible, but applying a boundary value problem solver the authors indeed find periodic solutions which are better than the best steady-state solution. An interesting paper.