Necessity of limiting co-state arcs in Bolza-type infinite horizon problem

Abstract

We investigate necessary conditions of optimality for the Bolza-type infinite horizon problem with free right end. The optimality is understood in the sense of weakly uniformly overtaking optimal control. No previous knowledge in the asymptotic behaviour of trajectories or adjoint variables is necessary. Following Seierstad’s idea, we obtain the necessary boundary condition at infinity in the form of a transversality condition for the maximum principle. Those transversality conditions may be expressed in the integral form through an Aseev–Kryazhimskii-type formulae for co-state arcs. The connection between these formulae and limiting gradients of pay-off function at infinity is identified; several conditions under which it is possible to explicitly specify the co-state arc through those Aseev–Kryazhimskii-type formulae are found. For infinite horizon problem of Bolza type, an example is given to clarify the use of the Aseev–Kryazhimskii formula as an explicit expression of the co-state arc.

Keywords:

• optimal control
• problem of Bolza type
• infinite horizon problem
• transversality condition for infinity
• uniformly overtaking optimal control
• limiting subdifferential
• unbounded cost
• shadow prices

AMS Subject Classifications:

• 49K15
• 49J52
• 91B62

Acknowledgements

I am grateful to the referee for precise remarks on the seemingly minuscule issues and would like to express my gratitude to Ya.V. Salii for the translation.

Notes

This study was supported by the Programme ‘Mathematical Theory of Control’ of the Russian Academy of Sciences [projects number 12-P-1-1019 and 12-P-1-1012], and by Act 211 Government of the Russian Federation No 02.A03.21.0006.