Krawczyk, Jacek B. and Serea, OanaSilvia (2007): A viability theory approach to a twostage optimal control problem.

PDF
MPRA_paper_10103.pdf Download (178Kb)  Preview 
Abstract
A twostage control problem is one, in which model parameters (“technology”) might be changed at some time. An optimal solution to utility maximisation for this class of problems needs to thus contain information on the time, at which the change will take place (0, finite or never) as well as the optimal control strategies before and after the change. For the change, or switch, to occur the “new technology” value function needs to dominate the “old technology” value function, after the switch. We charaterise the value function using the fact that its hypograph is a viability kernel of an auxiliary problem and study when the graphs can intersect and hence whether the switch can occur. Using this characterisation we analyse a technology switching problem.
Item Type:  MPRA Paper 

Original Title:  A viability theory approach to a twostage optimal control problem 
Language:  English 
Keywords:  value function, viability kernel, viscosity solutions 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C69  Other C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C61  Optimization Techniques; Programming Models; Dynamic Analysis 
Item ID:  10103 
Depositing User:  Jacek Krawczyk 
Date Deposited:  20. Aug 2008 03:27 
Last Modified:  16. Feb 2013 02:00 
References:  [1] AUBIN J.P. (1992) Viability Theory. Birkh¨auser. [2] AUBIN J.P. (2001) Viability kernels and capture basins of sets under differential inclusions, SIAM J. Control Optimization, Vol. 40, No.3, pp. 853881. [3] BARLES G. (1994) Solutions de viscosit´e des ´equations de HamiltonJacobi. Springer, Paris. [4] BARDI M. & CAPUZZO–DOLCETTA I. (1997) Optimal Control and Viscosity Solutions of HamiltonJacobiBellman Equations. Birkh¨auser, Boston. [5] BARRON E.N. & JENSEN R. (1990) Semicontinuous Viscosity Solutions of HamiltonJacobi Equations with Convex Hamiltonian, Comm. PDE., vol. 15, pp. 1713–1742. [6] Bene, C., L. Doyen & D. Gabay, (2001), A viability analysis for a bioeconomic model, Ecological Economics, 36, pp. 385–396. [7] CARDALIAGUET P., QUINCAMPOIX M. & SAINTPIERRE P. (1999) Set valued numerical analysis for optimal control and differential games, in "Stochastic and differential Games: Theory and Numerical Methods", Ann. Internat. Soc. Dynam. Games 4, Bardi M. , Raghavan T.E.S. and Parthasarathy T. , eds. , Birkhaeuser, Boston, MA, pp. 177–274. [8] CARDALIAGUET P., QUINCAMPOIX M. & SAINTPIERRE P. (1997) Optimal times for constrained nonlinear control problems without local controllability, Appl. Math. Optimization, Vol. 36, No.1, pp. 21–42. [9] CRANDALL M.G., ISHII H. & LIONS P.L. (1992) User’s guide to the viscosity solutions of HamiltonJacobi equations, Trans. Amer. Math. Soc., Vol. 282, pp. 487502. [10] DE LARA, M., L. DOYEN, T. GUILBAUD, M.J. ROCHET (2006), “Is a management framework based on spawning stock biomass indicators sustainable? A viability approach”, ICES Journal of Marine Science, in press; doi:10.1093/icesjms/fsm024 [11] EVANS L.C. (1998) Partial Differential Equations, Graduate studies in mathematics, Vol. 19, A.M.S., Providence, Rhode Island. [12] FRANKOWSKA H. (1993) Lower Semicontinuous Solutions of the Bellman Equation, SIAM J. Control Optim., Vol. 31, pp. 257–272. [13] KRAWCZYK, J.B. & KUNHONG KIM (2004), “A Viability Theory Analysis of a Simple Macroeconomic Model”, New Zealand Association of Economists Conference 30th June  2nd July 2004, Wellington, New Zealand, Conference Website. [14] KRAWCZYK, J.B. & KUNHONG KIM (2008), Satisficing Solutions to a Monetary Policy Problem: A Viability Theory Approach, Macroeconomic Dynamics, forthcoming. [15] KRAWCZYK, J.B. & R. SETHI (2007), Satisficing Solutions for New Zealand Monetary Policy, Reserve Bank of New Zealand Discussion Paper Series, No DP2007/03. Available at url: http://www.rbnz.govt.nz/research/discusspapers/dp07 03.pdf [16] MARTINET, V. & L. DOYEN (2007), Sustainability of an economy with an exhaustible resource: A viable control approach, Resource and Energy Economics, vol. 29(1), pages 1739. [17] Martinet V., O. Th´ebaud & L. Doyen (2007), Defining viable recovery paths toward sustainable fisheries, Ecological Economics, in press [doi:10.1016/j.ecolecon.2007.02.036] [18] CLEMENTPITIOT, H. & L. DOYEN (1999), Exchange rate dynamics, target zone and viability, Universite Paris X Nanterre, manustcript. [19] CLEMENTPITIOT, H. & P. SAINTPIERRE (2006), Goodwin’s models through viability analysis: some lights for contemporary political economics regulations, 12th International Conference on Computing in Economics and Finance, June 2006, Cyprus, Conference Maker. [20] PUJAL, D. & P. SAINTPIERRE (2006), Capture Basin Algorithm for Evaluating and Managing Complex Financial Instruments, 12th International Conference on Computing in Economics and Finance, June 2006, Cyprus, Conference Maker. [21] PLASKACZ S. & QUINCAMPOIX M. (2001) Value Function for Differential Games and Control Systems with Discontinuous Terminal Cost, SIAM J. Control Optim. Vol. 39, No. 5, pp. 14851498. [22] PLASKACZ S. & QUINCAMPOIX M. (2000) Discontinuous Mayer Problem Under StateConstraints, Topological Methods in Nonlinear Analysis, Vol. 15, pp. 91100. [23] PLASKACZ S. & QUINCAMPOIX M. (2002) On Representation Formulas for Hamilton Jacobi Equations related to Calculus of Variation Problems, Topological Methods in Nonlinear Analysis. [24] QUINCAMPOIX M. & VELIOV V. (1998) Viability with a target: Theory and Applications, in Applications of Mathematical Engineering, Cheshankov B. and Todorov M., eds. ,pp. 4758, Heron Press, Sofia. [25] SEREA O.S. (2001) Discontinuous Differential Games and Control Systems with Supremum Cost, Prepublication du Departement de Mathematiques de L’Universite de Bretagne Occidentale, Brest. [26] SUBBOTIN A. I. (1995) Generalized Solutions of FirstOrder PDEs. The Dynamical Optimization Perspective, Birkhaeuser, Boston. [27] TOMIYAMA, K. (1985), TwoStage Optimal Control Problems and Optimality Conditions, Journal of Economic Dynamics and Control, 9, 317337. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/10103 